Numerical modelling of the surf zone dynamics at Waimea Bay

Technical University of Denmark - Spring 2006
Numerical modeling of the surf zone dynamics in Waimea Bay
Masters of Science Project
Student: _______________________________________ Date ___________
S011250 Simon Brandi Mortensen
Supervisors:
Professor Per A. Madsen, MT, MEK, DTU
Professor Kwok Fai Cheung, University of Hawaii
Henrik Kofoed-Hansen, DHI Water & Environment
1
Acknowledgements
The University of Hawaii’s Department of Ocean and Resources Engineering has most kindly
hosted this research project under the supervision of Professor Kwok Fai Cheung. Professor Per
Madsen from MEK DTU and Henrik Kofoed-Hansen from the DHI have been assisting as
external supervisors. First of all I would like I would like to express my deepest gratitude to my
three supervisors for allowing me to undertake this research project and for your kind help and
assistance during the course of the study.
I would like to thank Professor Geno Pawlak for providing the necessary instruments and
equipment for the field campaign and for his counseling and friendly coaching, and also Professor
Mark Merrifield for his helpful advice on how to analyze the obtained field data.
Thank you to PhD student Marion Chavanne for teaching me how to calibrate the instruments and
my deepest thanks to PhD student Monte Hansen, Kimball Millikan and PhD Andrew B. Hebert
for obtaining the necessary permits and helping me carry out the underwater instrument
deployment.
Thanks to PhD student Volker Roeber for providing me the surveyed Waimea Bay bathymetry
data, to Professor Jannette Frandsen for lending me an extra computer for carrying out model
simulations and to Tyson Hilmer for teaching me about digital video processing.
Thank you so much for your kind help and assistance.
My warmest Aloha to all of you.
Simon B. Mortensen
Denmark, July 25. 2006
2
1
2
3
4
5
6
7
8
Introduction............................................................................................................................. 4
Governing Equations ............................................................................................................... 6
Preliminary Study.................................................................................................................. 13
3.1
Case 1 –Surf zone dynamics on a plane sloping beach ................................................... 13
3.1.1
Mild slope, spilling breaker:................................................................................... 13
3.1.2
Mild slope, spilling breaker.................................................................................... 21
3.1.3
Steep slope, plunging breaker................................................................................. 22
3.1.4
Propagation and breaking of irregular wave train ................................................... 23
3.2
Case 2 – 2D simulation of wave induced horizontal current circulations ........................ 25
3.2.1
A detached breakwater........................................................................................... 25
3.2.2
Rip current ............................................................................................................. 29
Waimea Bay.......................................................................................................................... 33
4.1
The Field Campaign ...................................................................................................... 33
4.1.1
Offshore boundary considerations .......................................................................... 33
4.1.2
Placing the instruments .......................................................................................... 35
4.1.3
The acquired field data........................................................................................... 37
4.1.4
Post-processing the field data. ................................................................................ 38
4.1.5
Selecting the MIKE21 BW input time-series. ......................................................... 43
4.2
The Model Setup ........................................................................................................... 47
4.2.1
Choosing the domain size ...................................................................................... 47
4.2.2
Generating the bathymetry ..................................................................................... 51
4.2.3
The sponge layer.................................................................................................... 55
4.2.4
Further discussion of the non-linear boundary conditions. ...................................... 57
4.2.5
Filter layer ............................................................................................................. 59
4.2.6
Porosity layer......................................................................................................... 59
4.2.7
Additional parameters ............................................................................................ 60
4.3
Simulation Results:........................................................................................................ 61
4.3.1
The first attempt..................................................................................................... 61
4.3.2
Introducing porosity layer ...................................................................................... 71
4.3.3
Extracting the reflected wave ................................................................................. 77
4.3.4
Increasing bottom friction ...................................................................................... 83
4.3.5
Enlarging the domain ............................................................................................. 87
4.3.6
Altering the wave direction .................................................................................... 93
4.3.7
Wednesday time-series (excl. porosity layer) ......................................................... 99
4.3.8
Wednesday time-series (incl. porosity layer) ........................................................ 105
Results Summary ................................................................................................................ 110
Conclusion .......................................................................................................................... 116
References........................................................................................................................... 120
Appendices.......................................................................................................................... 121
3
1 Introduction
On the north shore of the Hawaiian island of Oahu located at 2138’ W 15803’ lies Waimea bay.
During the winter months of November to March; Waimea Bay is frequently bombarded with long
period groundswell with wave periods of typically 15-22 sec and wave heights ranging from 2 m to
occasionally more than 10 m as measured by the Waimea wave buoy located 5.6 km offshore the
bay in a water depth of 200 m.
figure 1 – Waimea Bay, North end
figure 2 – Waimea Bay, South end
The bay faces WNW (300) and with the almost constant trade winds blowing from the east,
Waimea bay is most often entirely sheltered from locally generated short period wind waves and
therefore only subject to the long period groundswell usually originating from a direction of 300350.
The almost rectangular shaped bay is only 640 m wide and 290 m long and with the exception of a
reef situated along the north end of the bay and a few rocky outcrops along the south side of the
bay, the bathymetry is fairly regular and mildly sloping with a average bottom gradient of 0.04.
4
figure 3
The 2D-module of the MIKE21 BW software package developed by the Danish Hydraulic Institute
is a state-of-the-art numerical modeling tool capable of modeling waves and wave driven currents in
the coastal zone very accurately. The model uses the enhanced time-domain Boussinesq type
equations initially presented in Madsen and Sørensen (1992) and later extended into the surf and
swash zone as described in Madsen et al (1997a,b) and Sørensen et al (1998).
The main constraint of this enhanced Boussinesq model is that the maximum L0/h ratio should be
less than 0.5 and the bathymetry should be mildly sloping in order for the predictions to be reliable.
Additionally the computational model domain should not be too extensive (in terms of the number
of grid points) in order for the computational run-time to be within reasonable limits.
As a result it is clear that Waimea bay is an ideal location for assessing the performance of the
MIKE21 BW model in a full-scale natural wave-environment.
The goal of this research project has been to use this enhanced Boussinesq model in an attempt to
simulate the complex wave transformations and wave-driven current processes occurring in
Waimea bay. For verification the model results are compared with field measurements obtained
from a small-scale field campaign carried out as a part of this project, and the results of this
comparison are thoroughly discussed and analyzed.
5
2 Governing Equations
The following chapter is based on the work presented in Madsen & Sørensen (1992), Madsen et al.
(1997a) and Sørensen et al. (1998).
The 2D module of MIKE21 BW is based on the two dimensional Boussinesq equations with
enhanced linear dispersion characteristics in a depth integrated flux-formulation presented in
Madsen et al. (1997).
 P Q


0
t x y
P   P 2
 
t dx  d
(1.)
   PQ  R xx R xy


  

 gd
 x  x  0

y
x

 dy  d  x
y
Q   Q 2    PQ  R yy R xy

  
 

 gd
 y 
0

t dx  d  dy  d  y
x
y

(2.)
(3.)
In this formulation d is the total water depth and P and Q are the depth-integrated velocities.
The terms x and y are dispersive Boussinesq type terms derived under the mild-slope assumption
(ignoring higher order derivatives of h(x)) and enhanced to provide improved frequency dispersion
characteristics by adopting a Pade expansion scheme in the truncation of the velocity terms instead
of using Taylor expansion. These equations were first derived by Madsen and Sørensen (1992) and
are expressed by:
3
 3
1  2  3P
 3Q 

3  



 x   B   h  2 
 Bgh  3 
3   x t xyt 
xy 2

 x



h  1  2 P 1  2 Q
 2
 2 
h  1  2 Q
 2 

 h 

 2 Bgh 2  Bgh 2   h 
 Bgh
x  3 xt 6 yt
x
y 
y  6 xt
xy 
  3
1    3Q
 3 
3P 


  Bgh 3  3 
 y   B  h 2  2 
3   y t xyt 
yx 2 

 y
h  1  2 Q 1  2 P
 2
 2 
h  1  2 P
 2 




h 

 Bgh 2  2 Bgh 2   h 
 Bgh
y  3 yt 6 xt
x
y 
x  6 yt
xy 
(4.)
(5.)
Where B is given by 1/15.
Linear dispersion characteristics
A small 1D study has been carried out in order to assess the governing equations linear dispersion
characteristics. By only considering the x-direction, equation (1) can be simplified to:
6
 P

0
t x
6.
Considering non-breaking wave conditions and ignoring bottom friction, equation (2) can be
simplified to (by inserting equation (4)):
2P 
 2
 3 P h 
 3

P
0
 Bgh 3 3  ( B  13 )h 2 2   2 Bgh 2 2  13 h
 gh
xt 
x
t
x
x t x 
x
7.
By cross-differentiation and combination of (6) and (7) you get:
3
4
h  
 3
 2
 4
 2
2   
3  
2
1


gh
Bgh
B
h
B
h
Bgh
(
)
g
(
2
1
)
5








3
dx 2
dx 4
x 3 
xt 2
x 2 t 2 x  x
t 2
8.
The solution to this equation can be expressed by:
 ( x, t )  A( x )e i (t  ( x ))
9.
Where A is the wave amplitude,  is the angular frequency and  is the phase shift expressed by:

 k (x)
x
10.
By inserting (9) into (8) and assuming that A, h and k are only slowly varying functions of x, their
spatial derivatives can be ignored and the linear dispersion relation is obtained given by:
2 


3 gk 2 h  Bgk 4 h 3
3  k 2 h 2  3Bk 2 h 2
11.
The linear dispersion relation derived from linear wave theory is given by:
 stokes  g  k  tanh( kh)
2
12.
From (11) and (12) the following ratio between the two is obtained and subsequently illustrated in
figure 4.
2
 stokes
3  kh  (1  151  kh  )  coth(kh)
2
2

3  18
15  kh 
2
13.
7
figure 4
From figure 4 it is observed how the governing equations show excellent linear dispersion
characteristics up to a kh of about 1 after which is starts to grow increasingly inaccurate for larger
kh values. The recommended maximum kh value in a Mike21 BW simulation is 3.15 (DHI
Software 2004), which in that case would correspond to a deviation of linear dispersion
characteristics of 5.7%.
Wave breaking
The evolution equations have been extended to include the effects of wave breaking based on the
concept of surface rollers, which is adapted from the breaking characteristics of a spilling breaker.
The surface roller concept assumes that the effect of wave breaking can be modeled by imposing a
volume of water (a roller) on the wave front (from the moment of breaking) traveling with the wave
celerity c.
This change in the wave velocity profile leads to and excess in momentum, which is accounted for
by the terms Rxx, Ryy and Rxy given by:
2
2

P 
P 
Q 

Q  

( R xx , R xy , R yy ) 
 c x   ,  c x     c y  ,  c y   
d 
d 
d
1   / d  
d 
(14.)
Here  is the roller thickness and (cx, cy) are the wave/roller celerities.
In this model breaking occurs when the local wave steepness at the time tB exceeds the initial
breaker angle B. After initial breaking the limiting gradient tan will gradually decay over time
until reaching its terminal value of tan0. This decay is included in order to simulates the broken
waves gradually transformation into a bore in the inner surf zone. The decay function is given by:
8

t  tB
tan  t   tan  0  tan  B  tan  0   exp  ln 2

t1 2





(15.)
Where t1/2 is half time for the transition from B and B.
The toe of the roller is then defined as points satisfying the condition that the absolute value of the
gradient equals the instantaneous local value of tan and that the gradient is negative in the
direction of the wave propagation.
figure 5
In each toe point a tangent is imposed with the slope of tan and pointing in the opposite direction
of the wave propagation as observed in figure 5. The roller thickness  is then straight forwardly
defined as the water above this tangent multiplied with the shape factor f.
In nature wave breaking can be divided into 4 categories: Plunging, spilling, surging and collapsing.
The breaker type can be determined using the surf similarity parameter given by:
  tan 
L0
Hb
(16.)
Where  is the beach slope, L0 is the deepwater wavelength and Hb is the wave height at the break
point. From Nielsen (2003) a table of the range of each breaker type is given.
Surf similarity parameter
Breaker type
4<
Little or no breaking (Reflection)
2<<4
Collapsing or surging breaker
0.4<<2
Plunging Breaker
<0.4
Spilling Breaker
table 1
9
It should be clear that each of these 4 types of breaking has different energy dissipation
characteristics as well as visual appearance and behavior. As mentioned earlier the “roller concept”
has been developed on the assumption of spilling breakers, which might provide an unsatisfying
inaccurate prediction of energy dissipation if trying to simulate a surf zone containing one of the
other 3 breaker types.
Wave celerity
The original Boussinesq model presented by Madsen et al. (1992) the wave celerity is calculated
interactively from the local wave field. However as this feature can result in stability problems in
the MIKE21 BW 2D-module the celerity is instead obtained from Sørensen et al (1997) given by a
two dimensional expression of the linear formulation c  1.3 gh :
  


 c x   x 
1.3  gh
 

c 
2
 y        2   

  

 
 y   x   y 
(17.)
Here  is the surface elevation and h is the still water depth. This linear celerity formulation has
been shown to provide good results for regular waves, but it does not capture irregular wave
phenomena like deceleration of the primary waves due to wave down-rush of long waves in the
swash zone, which can be important in some model applications.
Bottom friction
The final terms in the evolution equations are the dispersive terms due to bottom friction which are
given by:

x , y  
P2  Q2
1
 fm   
 P, Q 
2
d2
(18.)
Here fm is the friction factor, P and Q the depth integrated velocities and d the instantaneous water
depth (d= h + ). However in the MIKE21 program interface the bottom selected in terms of the
Chezy number C where:
C
2g
fm
(19.)
10
Hence:
P2  Q2
1 2g
 x , y    2   
P, Q 
2 C
d2
(20.)
Moving shoreline
In order to be able to handle the moving boundary at the shore and gain the ability to simulate
swash zone properties such as the wave run-up, the numerical model uses a method typically
referred to as the “Slot technique” described in Madsen et al (1997). Using this approach the
computational domain is extended artificially by introducing an artificial porous flow domain on the
impermeable beach by introducing a permeable beach with very small porosity. Numerically the
permeable beach is treated like water filled “slots” allowing waves to “propagate” into the beach
slope and thereby handling the zero-water depth problem in the numerical calculations when trying
to calculate a point based on a water point and a land point. The transition between the physical and
porous flow regime is given by:
1

 ( z)  
 ( z  Z L ) /( Z L  Z B )
  (1   )  e
ZL  z
ZB  z  ZL
(21.)
Here (z) is the porosity,  is the minimum value of ,  is a constant shape factor defining the
exponential transition between the physical and porous flow regime, z is the vertical coordinate, Z L
is the vertical location of the physical seabed and ZB defines the lower limit of the porous region,
which as minimum should be located offshore the swash zone which marks the limit for the
fluctuation of the moving boundary.
The resulting effective water depth A can then be calculated as:

A( x, y, t )    ( z )dz
(22.)
ZB
Thus in the porous flow domain the water depth A is used in the depth-integrated momentum
equations (1) and (2) instead of d. However as the porosity domain consists of a porous beach with
clear water on top in the depth-integrated continuity equation can not be straight forwardly modified
to include a depth-integrated porosity as this would lead to porous flow everywhere in this domain.
Instead (3) is modified by:

 P Q


0
t x y
(23.)
11
Where:
 ( x, y , t )   ( )
(24.)
Where  is the surface elevation.
The numerical scheme
For the purposes of this study the differential equations used in the 2D-module of the MIKE21 BW
is discretisized using a time-centered implicit scheme with variables defined on a space-staggered
rectangular grid. The method is called the System21 scheme and is based on the Altering Direction
Algorithm. The resulting system of finite difference equations is reduced to a tri-diagonal system,
solved by the Double Sweep Algorithm as explained in Madsen (1992). However as the model is
known to become unstable in regions with steep gradient introducing artificial oscillations, a simple
unwinding scheme is applied in theses areas. The transition is set to occur when the absolute value
of the second-difference of the local depth-integrated velocity grows larger than the first-difference
of the local depth-integrated velocity.
When resolving the Boussinesq cross-terms the standard representation results in these terms to be
back centering half a time-step, which will result in an artificial dissipation of waves traveling with
an angle to the grid. As a result a linear extrapolation scheme has been used to avoid this effect.
However this method of time-centering the cross-terms will sometimes result in model instability,
which is accounted for in the model by incorporating a time-extrapolation factor, which can attain
values between 0 and 1. A value of 1 represents a correct time centering and 0 represents a
backward centering of the Boussinesq cross-terms.
12
3 Preliminary Study
Due to the extensive complexity of the modeling task at hand, a preliminary study has been carried
out, where the most governing wave transformation processes has been modeled separately on
simplified bathymetries using MIKE21 BW and the results compared with laboratory experiments.
The preliminary study has been divided into 2 case studies.

In case study 1 MIKE21 BW is used to model the cross-shore motion and breaking of
regular and irregular waves on a plane sloping beach.

In case study 2 MIKE21 BW is used to model the rip channel flow on an open beach
and the circulation flow behind a breakwater.
The primary objective of these case studies is to carry out a proper assessment of how well MIKE21
BW deals with modeling each of these wave-induced processes individually using simplified model
domains that allows the MIKE21 BW model predictions to be benchmarked with laboratory
experiments.
The model setup and verification for both test cases in the preliminary study has been based on the
work carried out in (Madsen et al. 1997 Part I-III).
3.1
Case 1 –Surf zone dynamics on a plane sloping beach
3.1.1 Mild slope, spilling breaker:
In the first part of this case study the wave-breaking module in MIKE21 BW is tested, by setting up
a model, where directional monochromatic waves are set to propagate over a gently sloping bottom
bathymetry.
As explained earlier the MIKE21 breaker module is based on the spilling breaker assumption.
For that reason it would be the most appropriate if the first model test in the assessment fulfilled
this criterion.
(Stive 1980) has carried out wave tank experiments with regular waves with an initial wave height
H of 0.145 m and a wave period T of 1.179 sec propagating on a beach slope of 1/40. The
deepwater water depth hmax is 0.7 m. Under these conditions a spilling breaker should be expected.
13
The model is set to simulate this experimental study using at grid spacing dx of 0.05 m. A 2nd order
stokes wave is set at the deepwater limit to represent the initial wave condition at the wave
generation line.
The time step dt is set to 0.005 sec, which for the given corresponds to a Courant number of 0.26.
Offshore the wave generation line a sponge layer is placed to damp out all wave oscillations
propagating offshore of it. The length of the sponge layer should be at least half the wavelength,
which for the given T and hmax results in a wavelength L of 4 m. For convenience the length of the
sponge layer Lsponge is set to 50 grid points (2.5 m) to assure that nothing is being reflected from the
offshore back wall. The wave generation line is located at the toe of the slope.
After reaching the shoreline the beach continues to rise to an elevation hBeach of 1.0 m, which
corresponds to a horizontal beach length LBeach of 40 m.
Figure 6
As explained earlier MIKE21 uses the slot technique to incorporate run-up, allowing water to
propagate into the slope. So while the expected horizontal run-up should not reach more than in the
order of 1 m, the unphysical propagation of water inside the slope extend much further. So the
straight forward reason for choosing such a large beach area is simply to be absolute certain that the
incoming waves do not get a chance to interfere with the back wall at the right boundary of the
domain creating numerical instabilities and inaccurate model predictions.
Near the initial shoreline a low-pass filter is applied in order to damp out high-frequency
instabilities and to dissipate the wave energy in the area where the surface roller cannot be resolved.
The simulation is run for 50.000 time steps (250 sec) in order to insure a proper development and
stabilization of the wave setup.
14
When outputting and spatially plotting the wave height and the mean surface elevation (wave
setup), it was quickly realizes that the position of the filter as well as the value of the filter
coefficient itself had a significant influence on the model output. It was also discovered that
different filter setups allowed for smaller slot widths than others and that the slot width itself also
had some influence on the on the model output.
Influence of the filter coefficient value.
Figure 7
In Figure 7 envelope plots are made for 3 simulations using each using a different filter setup. For
all 3 simulations a slot width b of 0.05 has been used.
The envelope plot to the left represents a simulation where no filter has been used. In the two other
simulations presented in the center and right side plot, a filter with a filter value of respectively 0.25
and 0.5 have been applied. Both filters are applied in the surf and swash zone and start at 21.25 m,
which is the first landward grid point following the determined breakpoint in the no-filter
simulation. It is noticed how high frequency high amplitude oscillations are appearing in the surf
and swash zone and how they are increasingly damped when the filter value is increased. With a
filter value of 0.5 these oscillations have almost been entirely damped in the surf zone and only
appear moderately in the swash zone close to the shoreline.
However an interesting observation was that the high-frequency high-amplitude oscillations in the
surf zone only appear during the first couple of time steps. When plotting the last 20000 time steps
of the 3 simulations, the envelope plots suddenly looks much more identical.
15
Figure 8
Gone are the high-frequency oscillations in the surf zone and now only the ones in the swash zone
remains for the no-filter case and also to a minor degree for the case with a filter value of 0.25.
The conclusion is that the high-frequency high amplitude oscillations in the surf zone must be
numerical instabilities that occur only during the first time steps until the model has reached quasistatic conditions, with a fully developed wave setup. As a result the high-amplitude instabilities in
the surf zone does not contribute significantly to a distortion of the evolution of the outputted phase
averaged wave height and setup in this region. But because the numerical instabilities can lead to
model “blow-up” and e.g. limits the minimum slot width that can be used in the model, it is
concluded to be the best approach to extend the filter into the surf zone.
Figure 9
But meanwhile the lifespan of the numerical instabilities in the surf zone are short, it is noticed that
high-frequency oscillations continue to exist in the swash zone near the shoreline, where the surface
16
roller cannot be resolved. These can only be removed by applying the low-pass filter and from
Figure 8 it can be concluded that the filter value should be 0.5 in order for the high-frequency
oscillations to be removed correctly. When comparing to (Stive 1980) it can be observed how the
model predicted wave height in the surf zone is overestimated if no filter is applied. The best wave
height model prediction is obtained if a filter value of 0.5 is used. Also when it comes to modeling
the wave setup a filter with a filter value of 0.5 provides the most accurate prediction of the
maximum wave setup at the shoreline. However on the other hand the “no filter” model simulation
seems to give a better prediction of the point of initiation of wave setup.
It is observed that the model predicts the wave breakpoint to occur to early when compared to
(Stive 1980). In the simulation where no filter is applied the difference is 0.9 m and when a filter is
used the difference is 1.1 m. The model predicted breakpoint can be shifted landwards by slightly
increasing the initial breaking angle, but it would not improve the overall wave height prediction of
the model.
The model predicts the horizontal shift between the breakpoint and the initiation of wave setup,
which is in (Madsen 1997) explained as being caused as result of the initial conversion of potential
energy into forward momentum flux.
The shift is 0.2 m for the simulations where a filter is applied and 0.85 m when no filter is used.
From (Stive 1980) the horizontal shift is measured to approximately 1 m, so it could be suggested
that the model provides better predictions if the filter is applied sufficiently landward of the
breakpoint.
Influence of the filter position
Next step is to investigate how important the seaward extension of the low-pass filter into the surf
zone is to model predictions. 3 model simulations with different offshore extensions of the filter
have been carried out and are compared to the simulation, where no filter is used.
For all simulations the slot width is 0.05 and the filter value is 0.5. In the simulation where no filter
is applied the wave breakpoint is noticed to be located at 21.2 m from the offshore back wall of the
model.
17
figure 10
figure 11
It is clearly observed in figure 10 that if the filter is extended offshore (at 20 m) beyond the
originally predicted breakpoint, it will lead to the breakpoint being predicted to be located at the
new filter position, which of course is not correct. Locating the offshore limit of the filter landward
of the breakpoint (at 23 m) does on the other hand not cause the predicted breakpoint to change
(compared to the no-filter case).
However it is interesting to notice that when applying the filter just landward of the original
breakpoint (occurring at 21.2 m), the new predicted breakpoint actually moves 0.25 m offshore.
From figure 11 it is observed how the placement of the filter also strongly affects the starting
location of the wave setup. When the filter is applied offshore the original breakpoint the location of
the wave setup also moves offshore corresponding to the original starting location (where no filter
is uses). When the filter is applied just landward of the original breakpoint the breakpoint moves
slightly offshore and so does the starting location of the wave setup. Applying the filter further
landward from the original breakpoint (at 23 m) does not change the starting location of the wave
setup compared with the no-filter simulation.
Finally it is noticed how the starting location of the wave setup is horizontally shifted landwards
compared to the respective wave breakpoint for all 4 simulations.
For the simulation where the filter is applied at 20 m, the shift is 0.15 m. When the filter is applied
at 21.25 m just landward of the original breakpoint, the shift is 0.2 and for the simulation with the
filter starting at 23 m the shift is 0.85 m just as when no filter is applied.
It is clear that the offshore extension of the filter has a major influence on how large this horizontal
shift is as predicted by the model.
18
When compared to (Stive 1980), the best wave height model prediction is obtained with a filter
located at 21.25m just landward of the break point. When the filter is applied offshore the original
breakpoint the model breakpoint prediction becomes completely wrong significantly decreasing
model accuracy. If the filter is not applied, the wave height in the swash and inner surf zone is
predicted inaccurately and the model easily becomes unstable. If the filter is applied at 23 m the
breakpoint remains unchanged at 21.2 m. The wave height for this simulation is predicted fairly
well with the exception of the area just landward of the breakpoint, where no filter is present.
Like in the previous study the initiation of wave setup is less well predicted if the filter is applied
immediately landward of the breakpoint but on the other hand provides the most accurate prediction
of the maximum wave setup near the shoreline. If the filter is applied at 23 m (1.8 m landward of
the breakpoint) the point of initiation is the same as when no filter is used, but the prediction of the
maximum setup is more accurate near the shoreline compared with the no filter case. If the filter is
applied offshore the original breakpoint, the initiation of wave setup is predicted inaccurately
offshore, and even though its prediction of maximum setup is the most accurate in the study
compared to (Stive 1980), this is not interpreted as a suggestion of good model performance, but
rather due to improper damping by the incorrectly placed filter.
From this study it can be concluded that the overall best model performance is obtained if the filter
is placed sufficiently landward of the original breakpoint. The filter should cover the swash zone
and not be extended further into the surf zone than strictly necessary to assure a stable model run.
Influence of the slot width
If for some reason the filter is not applied or if using a smaller filter value, this poses a lower limit
to the allowed slot width in order to avoid blow-ups. 3 simulations using the same filter starting at
21.25 m and a filter value of 0.5 has been carried out and the results presented in Figure 12.
19
Figure 12
Figure 13
It is noticed how a slightly better prediction of the wave height if a slot width of 0.01 or smaller is
used. However the model predicted magnitude of the wave setup becomes larger and deviates
further from the experimental results by (Stive 1980) compared to the simulation where the slot
width is 0.05.
The explanation is that as the water filled slots in the slope are increased; more water is allowed to
propagate and be “lost” in the slope making the system less physical, which e.g. results in an
underestimation of the magnitude of wave setup, even though this prediction is closer to the
experimental results. There seem to be no difference between model results carried out with a slot
width of 0.01 and 0.005. It is recognized that a smaller slot width can lead to model instability and
eventually blow-up, but it should always be attempted to use a small a slot width as possible in
order to make the computational domain (and thereby model predictions) as physical as possible.
20
3.1.2 Mild slope, spilling breaker
Another quite similar test case has been carried out in order to compare model performance with
laboratory experiments carried out by Hansen & Svendsen (1984). The model setup consists of a
rectangular flume with a horizontal section with a water depth of 0.36 m followed by a plane
sloping beach of 1/34.25. The input wave condition uses is a monochromatic wave with a period of
2.0 seconds and a wave height of 0.12 m. Calculating the surf similarity parameter yields a spilling
breaker in the flume. However it is noticed that the applied wave condition yields a highly nonlinear solution at the model offshore boundary and as a result a Cnoidal wave has been used as input
in the model.
figure 14
figure 15
From figure 14 and figure 15 it is again confirmed that the best agreement with laboratory data is
obtained when applying a filter in the swash zone and how a misplaced filter near the actual break
point results in an inaccurate prediction of the break point and the initiation of setup. As a result
only the simulation containing a filter applied in the swash zone (the red line) will be commented in
the following paragraph.
For this simulation it is noticed that the model predicts the initial shoaling very well in agreement
with Hansen & Svendsen (1984) and also the surf zone is well predicted by the model. However it
is noticed that the shoaling just offshore of the breakpoint is notably underestimated by the model,
which is assumed to be a result of the models limited capability in handling non-linear shoaling.
Non-linear shoaling is a combination of linear shoaling and an energy transfer to higher order
harmonics. The Boussinesq model used in this study has proven to have excellent linear shoaling
characteristics up to a kh of 3.15 for the primary wave harmonic but due to the same kh-limitation it
21
generally underestimate the magnitude of the higher order harmonics. As an example the 2nd order
harmonic is underestimated with more than 10% for a kh larger than 0.5. In comparison the kh for
this simulation is 1.1 near the breakpoint and as so the underestimated shoaling at this location
would be expected.
3.1.3 Steep slope, plunging breaker
Yet another simulation has been carried out to test the models capability in modeling a surf zone
containing plunging breakers. As recalled the breaker module in the model was originally designed
to deal with spilling breakers, but as a plunging breaker is expected to be dominant when modeling
the surf zone dynamics in Waimea Bay, it was found important to pre-assess the model performance
in such conditions.
figure 16
A model simulation has been set up based on a laboratory experiment conducted by Ting & Kirby
(1994) in a rectangular flume with a water depth of 0.4 m followed by a beach slope of 1/35. The
wave conditions consist of monochromatic waves with period of 5.0 sec and a wave height of 0.125
m. In the model Cnoidal waves has been used to provide the offshore input. The grid spacing and
time step is the same as for previous simulations. In order to account for the spilling breaker the
initial breaker angle has been adjusted from 20 to 25 and the half-time cut-off from 1 sec (T/5) to
0.5 sec (T/10). The surface roller shape factor has been set to 2.
From figure 16 it is noticed how the break point is very well predicted and also the non-linear
shoaling characteristics are much better compared to figure 14 in the previous simulation. The
22
explanation is that the wave numbers are much smaller for this simulation (khmax=0.26) thus the
higher order harmonics near the break point are much better estimated compared to the previous
simulation. Overall the agreement between Ting & Kirby (1994) and model results are most
satisfying and contributes to the overall justification for using the present model to simulate the surf
zone dynamic in Waimea Bay.
3.1.4 Propagation and breaking of irregular wave train
The input wave time-series used in the Waimea Bay model simulation is obtained from a measured
surface elevation time-series provided by an ADCP placed at the location of the model offshore
boundary. The resulting input wave time-series in the model will resemble an irregular wave train
that will go through a large number of complex transformation processes while propagating through
the domain.
Due to the complexity of the 2D-modeling task at hand, it was found reasonable to first assess
Mike21 BWs capability to simulate some of these processes (such as triad-interactions and bound
harmonic releases due to breaking of irregular wave-trains) in a 1D environment using a simplified
bathymetry.
In a laboratory study by Mase (1994) a bichromatic input was applied in a rectangular wave flume
consisting of a flat section with a water depth of 0.47 m followed by a sloping beach of 1/23. In the
experiment the bichromatic input were all given expressed by:
  a  cos2  f1t   a  cos2  f 2 t 
Where
 
f 1  1    f m
2

 
f 2  1    f m
2

figure 17
23
Along the beach slope 12 wave gauges were positioned to measure the surface elevation time-series
at each of these respective locations. The model was started at WG1 using the measured time series
at this location to generate the model input condition. Subsequently the predicted time-series at WG
8, WG 10 and WG12 was compared with laboratory experiments. The experimental test case used
for the comparison had an input bichromatic wave condition using =0.2 (WP2) and fm = 0.3 Hz,
which would result in plunging breakers in the model.
figure 18
figure 19
figure 20
From figure 18, figure 19 and figure 20 it is observed how a good agreement between experimental
data and model results are maintained throughout all 3 benchmarking locations, which indicates that
processes such as non-linear energy transfer to bound higher order harmonics and dissipation of the
irregular wave train due to breaking is well simulated by the model.
24
3.2
Case 2 – 2D simulation of wave induced horizontal current
circulations
3.2.1 A detached breakwater
In this section a 2D simulation modeling the wave transformation processes and wave induced
current circulation around a detached breakwater has been carried out. The model setup builds on a
laboratory study originally carried out by Hamm et al. (1995) and consists of 30x30 m basin starting
off with a 4.4 m wide horizontal section with a water depth of 0.33 m followed by a sloping beach
with a gradient of 1/50 until reaching the shoreline. From here the emerged beach has a slope of
1/20. Located 9.33 m from the shoreline a detached breakwater is placed being 6.66 m long and
0.87 m wide. The system is exposed to a monochromatic wave input with wave period of 1.69 sec
and a wave height of 0.08 m.
figure 21
The wave basin has been used to generate the model domain using a grid size of 0.05 m and a time
step of 0.02 sec. Behind the wave generation line a 2.5 m wide sponge layer (50 grid points) has
been applied to prevent wave reflection from the offshore back wall. A porosity layer enclosed on 3
sides by an impermeable wall is used to simulate partial reflection from the detached breakwater.
Also a porosity layer at the upper boundary has been used to damp out eventual reflection and
prevent instabilities in this area. The reflection coefficient used for this is 0.85. The breaker module
has been set to simulate spilling breakers and the moving shoreline module uses a slot depth of 0.33
25
m and a slot width of 0.02. A filter is applied in the swash zone extending out to a water depth of
0.067 m. The filter coefficient used is 0.5 as for previous simulations.
The bottom friction factor used in the model was given by a Chezy number of 65 corresponding to a
fm of 0.005.
figure 22
A Boussinesq wave input has been applied at the offshore wave generation line and the simulation
is carried out for 80000 time-steps (947 wave periods) and the phase-averaged properties (currents,
wave heights) are calculated on the basis of the last 10000 time-steps (118 wave periods).
Subsequently the wave height profiles behind the breakwater are compared to the laboratory results
at 4 locations. Also a comparison of the circulation current behind the breakwater is carried out.
26
figure 23
From figure 23 a decent agreement between model results and laboratory experiments is observed.
Due to the significant velocity gradients near the breakwater it was found that simulation wave
height predictions were quite sensitive to a variation in bottom friction. Also the way the partial
reflection was handled at the offshore side of the breakwater and at the upper model boundary
showed to have some minor influence.
27
figure 24 – circulation current, Hamm (1995)
figure 25 – Circulation current, Mike21 BW
From the circulation current comparison it is observed how there exists an excellent agreement
between the current circulation plot predicted by the model and the one measured by Hamm (1995)
as the only notable difference is that the circulation structure appear to be extending slightly further
out from the breakwater in comparison with laboratory measurements. Notice how the predicted
return current (near y= 16 m) causes small inundations in the breaker fronts in figure 22 as the
opposing current causes result in a wave steeping and thereby an offshore shift of the breakpoint at
this location.
28
3.2.2 Rip current
In a laboratory experiment originally carried out by Hamm (1992) and presented in Sørensen et al.
(1998), a 30 x 30 m wave basin has been used to simulate the rip-current generation on an open
beach.
In the last 2D-simulation in the preliminary study, a simulation has been carried for this setup and
the results compared to the experimental data.
Figure 26
29
The model domain consists of a 7 m horizontal section with a water depth of 0.5 m followed by a
sloping beach of 1/30. A small extravagation is made in the middle of the beach to represent the
typical bottom feature of a rip channel. The grid size is 0.05 m and the time-step is 0.01 sec. The
wave input is monochromatic with wave period of 1.25 sec and a wave height of 0.07 m. behind the
wave generation line a 2.5 m sponge has been applied to prevent reflection from the offshore
boundary. A filter has been applied in the swash zone extending out to a water depth of 0.035 m
using a filter value of 0.5. In the moving shoreline module a slot depth of 0.5 m and a slot width of
0.05 are used and the wave breaker module is set to simulate spilling breaking. In order for this
simulation to remain stabile it was found necessary to reduce the time-extrapolation factor to 0.5.
The bottom friction was set to a Chezy number of 35 corresponding to a fm of 0.016.
The simulation was carried out for 80000 time-steps (640 wave periods) and the wave height
distribution and wave-induced currents were calculated on the basis of the last 10000 time-steps (80
wave periods).
figure 27
figure 27 shows a good agreement between predicted wave height and model results. However it is
acknowledged that the break point is predicted to occur slightly landward of what was experienced
in the experiments by Hamm (1992). The explanation is expected to be due to small disagreements
in domain bathymetry compared to the laboratory test case.
30
figure 28 – Sørensen et al. (1998)
figure 29 – Current model simulation
The laboratory study by Hamm (1992) did not include a survey of the generated rip current so
instead the rip current obtained in this study (figure 29) has been compared with Sørensen (1998)
that previously used the Boussinesq model to simulate the same case study, which is presented in
figure 28. From the two figures above it is observed that the predicted velocities are with in the
same magnitude and the overall agreement is very good.
figure 30
31
However from figure 30 it is observed how the predicted rip current profile is slightly asymmetrical
in spite of the symmetrical model domain. Apart from suggesting that this might be the result of
small instabilities in the numerical computations no definitive explanation for this phenomenon
could be given in this paper.
32
4 Waimea Bay
4.1
The Field Campaign
In order to provide real case offshore boundary conditions for the Waimea Bay model simulation
and in order to be able to calibrate and assess the model predictions inside the bay, a small-scale
field campaign was initiated.
The simple purpose of this field campaign was to gather simultaneous time-series data of the
surface elevation and wave driven velocities inside the bay and at a suitable offshore location
during an event of long-period wave conditions (pop: a swell event).
4.1.1 Offshore boundary considerations
The first important decision was to be made on where to put the instrument located offshore of the
bay. As the wave time-series data gathered at this location will define the offshore boundary
condition for the model, two important conditions have to be satisfied.

The kh-ratio as this location should not exceed 3.15 in order to keep the linear dispersion
characteristics of the enhanced Boussinesq equations within their limit of accuracy (DHI
Software 2004). Here k being the wave number and h the still water depth.

When dealing with an irregular wave time-series as offshore boundary condition in the
MIKE21 BW model, the surface elevation signal is decomposed into all its harmonic
components, where each will be sent into the computational domain as free waves. As a
result the bound higher-order harmonics in a non-linear signal will instead be treated as free
waves causing boundary conditions to be resolved incorrectly. As a result the location of
the offshore boundary should be located in sufficiently deep water where the wave condition
for the desired swell event can be assumed to be linear.
As the desired offshore wave spectrum is expected to be narrow banded (no presence of short
period wind-waves) with wave periods ranging from 13-20 sec, the constraint of the possible
offshore water depth given by the kh-limit is by far exceeded by the limit of the maximum size of
the domain (number of grid-points) and the depth limitations of the instruments. Considering the
conservatively expected minimum wave-period to be 10 seconds, the model allowed deep-water
limit is as deep as 78 m. However the number of grid-points required extending the model from the
33
shoreline into a water depth this deep would make computational run-time unrealistically large not
to mention the practical difficulties associated with mounting an instrument at this location.
In the end it is the instrument available for deployment that sets the deepwater limit for the offshore
boundary. The 1200kHz ADCP Workhorse Sentinel (Acoustic Deepwater Current Profiler) made
available for this field campaign is a bottom-mounted instrument that has a max absolute water
depth limit (d = h + ) of 25 m. The ADCP is produced by Teledyne RD Instruments.
As large waves are common at this location during a regular swell event the water depth of the
ADCP deployment was chosen to be 21 m in order to assure that the instruments depth limitation
would not be an issue during sampling.
With the depth of offshore boundary decided, it was investigated if linear wave conditions could be
assumed at this location. Linear wave conditions can be assumed if following two conditions are
satisfied (Nielsen 2003).
H  L2
 1 (Ursell's parameter)
d3
(25.)
and
H/L <<1
(wave steepness ratio)
(26.)
The wave heights H experienced in a typical swell event ranges between 1 to 4 m with periods
ranging from 10-20 seconds corresponding to wavelengths (at h = 21 m) between 156-277 m.
As observed the wave steepness condition is satisfied easily, but unfortunately this is not the case
for the criterion posed by the Ursell’s number that for these wave conditions ranges from 2.62 to
more than 33.
As a result it is clear that it would not be possible to achieve pure linear conditions at the offshore
boundary as even the best case scenario still would be weakly non-linear.
With this information in mind it was decided to try to launch the field campaign for a small sized
swell event assuming that the non-linearities at the offshore boundary could be considered as weak
and insignificant not resulting in significant distortion of the model results. Additionally it was
decided that the duration of instrument deployment should be at least1-2 days in order to assure that
at least somewhere within this time span, the wave conditions would satisfy the condition of being
only weakly non-linear.
34
4.1.2 Placing the instruments
In addition to the ADCP placed at the model offshore boundary, two Nortek Current Profilers
(AquaDopps/AQDPs) has been assigned to measure the wave conditions inside the bay. Due to the
very limited number of instruments available for this study it was decided to place all three
instruments along the bay-normal centerline and focus on capturing the wave transformations
occurring from offshore into shallow water along this line. As mentioned earlier the ADCP is
deployed offshore in 21 m of water. The first AQDP is deployed at the bay entry in a water depth of
10.7 m and the last AQDP is placed approximately in bay middle in 7.4 m of water. These 3
locations will from now on be addressed Station 1, 2 and 3.
An overview of the instrument deployment is illustrated in figure 31:
figure 31
Instrument
Latitude
Longitude
ADCP
AQDP 1 (Bay Entry)
AQDP 2 (Bay Middle)
N 21.64356
N 21.64134
N 21.64069
W 158.07115 21 m
W 158.06622 10.7 m
W 158.06508 7.4 m
Mean Water Depth
35
table 2
From figure 31 it is quickly noticed that the location of the ADCP is not entirely in-line with the
two Aqua Dopplers. The simple explanation for this is a drifting boat anchor during deployment of
ADCP that caused the instrument in being placed slightly downwind of the planned location.
However this small displacement is of course not expected to be of any relevance whatsoever.
Above: Instruments ready for deployment
The ADCP and AQDP 1 were each mounted on an angular iron that was pressure drilled
approximately 1.5 meters vertically into the sand bottom to keep them stable and slightly elevated
above the sea bed in order to prevent eventual burial by moving sand. The angular irons were made
of aluminum in order not to disturb the instruments internal compasses and were equipment with
flaps to prevent them from rotating after deployment. AQDP 2 was designed to be mounted
horizontally and was fixed to two aluminum rods that were pressure drilled into the sand bottom.
Pressure-drilling the aluminum rods into the bottom.
Mounting AQDP 2 in 7.4 m of water.
36
During the 48 hours of deployment no notable migration or burial of any of the instruments were
experienced.
4.1.3 The acquired field data
Due to busy department schedules and the time required obtain the necessary permits for deploying
instruments in Waimea bay, the field campaign was originally scheduled to be carried out in the
beginning of March 2006. But in this month the island of Oahu experienced the worst series of
flooding events in 30 year, which caused septic tanks on the north shore to overflow and spill into
the ocean causing health hazardous conditions (for the divers) in Waimea Bay for more than 40 day.
Final diver preparations before deployment of AQDP 2 in Drifting anchor problems during the offshore deployment
the bay middle.
of the ADCP.
By the time conditions in the bay had turned to normal in April the chances of a decent swell event
(that normally only occurs during the winter months) had become very slim. But luckily a late
season swell suddenly showed up on the weather charts so that on Tuesday April 18, the field
campaign could be initiated and the instruments deployed in calm oceans and timed to start
sampling on Wednesday morning where a the swell was predicted to arrive.
The instrument started measuring at 9 am Wednesday morning and collected wave and current data
for the following 48 hours. All three instruments was set to their maximum sampling frequency of 1
Hz allowing for a 10-20 point resolution of each wave, which was considered to be satisfactory. The
data collected by all 3 instruments were stored in binary files from which the hydrodynamic
pressure at the instrument location at the bottom and the two depth-averaged velocity components
could be extracted using the bundled software from the instrument manufacturers.
37
4.1.4 Post-processing the field data.
The surface elevation is outputted in terms of a pressure time-series in dBar (10-1 bar) measured by
a sensor located on each of the 3 bottom-mounted instruments.
figure 32
figure 33
Using the block-averaging scheme (block size: 30 minutes) the fluctuations due to tidal motion is
removed from the raw pressure time-series and the oscillating pressure component p+ extracted
from the signal for further processing.
For a linear wave the relation between the surface elevation  and p+ at an arbitrary location in the
water column is given by:
cosh k i  h
1
cosh k  z 
p  ( z, t)
  (t ) 
 G N (k i ) 

g
  g cosh k i  z1 
cosh kh
(27.)
Here h is the positive still-water depth (measured from the seabed), z1 is the positive distance from
the pressure sensor to the seabed, GN is the linear transfer function and ki is the wave number
obtained from the linear dispersion relation expressed by:
2  f i 2  g  k i  tanh( k i  h)
(28.)
In order to calculate the actual surface elevation using this equation, a Fourier transform (FFT) of
the p+ time-series is used to decompose the signal into harmonic components for which the linear
transfer function can be applied to calculate the surface elevation contribution from each respective
harmonic. Finally the resulting surface elevation time series is obtained by merging the individually
modified harmonics.
N
 (t )  a 0   a n  cos  f n  2  t   bn  sin  f n  2  t   G N ( k n )
(29.)
n 1
38
Here an and bn are the real and imaginary components of the complex amplitudes obtained from the
Fourier transform and fn is the corresponding frequency ranging from 1/fNy to fNy, where fNy is the
Nyquist frequency given by 0.5 Hz (half the sampling rate of 1 Hz).
Special consideration had to be given on where to apply the frequency cutoff corresponding to the
parameter N. GN is an exponential growing function and for larger values of fn it will lead to
artificial amplification of high-frequency noise existing in this end of the frequency domain. As a
result a reasonable frequency cut-off must be introduced. With the model kh limit of 3.15 the
minimum wave period that can be resolved correctly at the 3 instrument respective locations
(starting from offshore) are Tmin={5.2 sec, 3.7 sec, 3.1 sec}. In this study it was decided to use the
model limitation for the frequency cutoff. Hence, fcut-off ={0.19 Hz, 0.27 Hz, 0.32 Hz} for the 3
locations.
figure 34
In figure 34 it is observed how the transfer function amplifies contributions from the higher order
harmonics revealing a wave train significantly more irregular in shape than what visually could be
observed from the raw pressure time-series. However it should be made clear that effects such as
instrument noise and not to mention the mathematical difficulties associated with resolving the
finite nature of the measured time-series always provide some contribution to the harmonic
amplitudes. While these effects are hardly felt in the lower range of the frequency domain, they
hold the potential to significantly distort the signal by even just small contributions to the small
harmonic amplitudes for the higher frequencies close to the cut-off (Smith 1999). As a result a
careful assessment has been made of each spectrum in order to assure that the defined frequency
39
cut-off for the 3 locations were acceptable estimates and only included frequencies that was part of
the wave signal.
The spectral wave peak period Tp is straightforwardly determined as the maximum harmonic
amplitude occurring in the wave-spectrum and is as such a measure for at what frequency that
contains the most energy. In order to obtain a distinguishable spectral distribution while still
keeping a fine frequency resolution, the frequencies are stored into bins of each 0.01 Hz wide, while
the FFT size is set to 4096 (using zero padding) corresponding to a frequency resolution f of
2.4410-4 Hz.
The integral wave parameters (Hmean, Hsig, Hmax) and the wave peak period can now be calculated
for the 48-hour field campaign. All these are calculated in 1-hour segments starting Wednesday
morning at 9 am. The integral wave parameters have been calculated using the zero-upcrossing
method.
figure 35
figure 36
On figure 35 it is observed how the size of the waves builds through Wednesday reaching its peak
late Thursday morning and then slowly declining in size until Thursday evening after which the
wave size remains constant through to Friday morning where the Field Campaign ended. Upon the
initial arrival of the swell event Wednesday morning the spectral peak wave period is 18.2 seconds,
which drops to 15.4 on Wednesday evening and finally to 13.3 sec Thursday evening. The decaying
trend in wave-period is a typical pattern for a swell event and is very common in Hawaiian waters.
The swells reaching the islands are most often generated thousands of kilometer away by powerful
40
storms generating waves with a broad spectrum of wave periods. As given by linear wave theory
the speed of wave in deep water will depend entirely on its period as given by:
c0 
g T
2 
(30.)
As a result the fraction of the generated waves with a longer period will travel faster and arrive
earlier than as a result a swell event in Hawaii (lasting typically 2-5 days) would usually begin with
long period forerunners and subsequently experience a downward trend in wave period until the end
of the swell event.
The mean wave directions (also from 1 hour intervals) are calculated from the two velocity
components measured by the ADCP.
figure 37
As the monitored swell origins from a storm thousands of kilometers away, the dominant wave
direction usually only changes a very few degrees during this event. From figure 37 it is observed
how the mean wave direction is 325 degrees on Wednesday morning changing to 327 degrees
Wednesday evening.
41
figure 38
figure 39
Finally the wave statistics calculated for the ADCP is compared with the concurrent wave statistic
taken at the Waimea buoy located 5.6 km northwest of the ADCP in 200 meters of water. The buoy
data is publicly accessible and can be found on the Coastal Data Information Program’s home page
(see references).
It is observed that there exists a very good agreement between the wave peak periods and the trend
in significant wave height is also in good in good agreement, with the one calculated for the ADCP.
The explanation for the significant drop in wave height between the two locations, could were well
be due to extensive refraction between the two locations.
42
figure 40
From figure 40 it is shown how the dominant wave direction at the offshore wave buoy 5.6 km
offshore is 350, while it is 327 at the location of the ADCP 540 m offshore the Waimea Bay
entry. As reference the shore normal of this part of the coastline is facing approximately 300 
4.1.5 Selecting the MIKE21 BW input time-series.
Due to extensive model run-times it was decided only to apply the MIKE21 BW model to 30
minutes of field data at the time. Instead of trying to simulate the entire 48 swell event focus has
been made on modeling only few discrete “slices”.
As mentioned earlier a potential problem with the model setup is that linear wave conditions at the
boundary are not fully satisfied. The least non-linear conditions (considering Ur) occur during
Wednesday and the most non-linear conditions occur Thursday morning.
In order to qualitatively assist model calibration, a digital camcorder has been used to film a limited
number of 5-30 min video-clips of the wave action in Waimea bay during the 48-hour deployment.
Due to limited camcorder battery life, only a few video clips where taken each day and only some
of these video clips contains easily recognizable occurrences such as the sudden arrival of a wave
much larger than the rest and also observable wave reflection.
Based on the considerations above, the two following 30-minute periods of the field data has been
selected for being modeled in MIKE21 BW:
43

16.15-16.45 Wednesday April 19. 2006

9.25 -9.55 Thursday April 20. 2006
The two time-series represents respectively the least non-linear and most non-linear wave
conditions during the field campaign and both contain video footage well suitable for visual
comparison.
An overview of the two input time series acquired from the ADCP is given below:
figure 41
figure 42
The power spectra generated for these time-series are presented in figure 43 using:
En 


1 2
2
a n  bn  f
4
1
(31.)
44
figure 43
From the spectral energy an expression for the significant wave height can be obtained using:
N
H m0  4  M 0  4  2   En 
n 1
4
2

 a
N
n 1
2
n
 bn
2

(32.)
The average wave direction for the two time series is obtained from scatter-plots of the horizontal
velocities:
figure 44
figure 45
From figure 41 and figure 42 the mean wave height Hm, the significant wave height Hsig and the
maximum wave height Hmax are obtained from the time-series using the zero-upcrossing method.
Finally a schematic overview of all the calculated parameters is presented in table 3.
45
Date
Hm
Hsig
Hmax
Tp
Hm0
Wave dir.
Ur (Hmax)
19/04-06 16.15-16:45
0.48 m
0.73 m
1.10 m
18.2 sec
0.64 m
325
7.4
20/04-06 9.25-9:55
1.14 m
1.79 m
3.06 m
15.4 sec
1.57 m
327
14.5
table 3
As mentioned earlier there is a significant difference between the 2 selected time-series, while in
spite of them being only 17 hours apart and belonging to the same swell event. While the wave
direction is almost the same there is a tremendous difference in wave height and spectral energy and
also a notable difference in wave peak period. From calculating the Ursell parameter using Tp and
Hmax (h = 21m) it is also confirmed (from previous discussion) that there is a significant difference
in non-linearity between these time-series both used as offshore boundary condition in the model.
46
4.2
The Model Setup
4.2.1 Choosing the domain size
The first concern when setting up the MIKE21 BW model is on what grid size (dx, dy) and timestep dt that should be used given the wave conditions at the offshore boundary and the expected
coastal dynamical processes that is expected to occur within the computational domain.
As the computational domain will include both wave breaking and a moving shoreline, the spatial
resolution of the wavelength of the most energetic wave should be 20-40 grid-points (Dhi Software
2004). Using a grid spacing of 2 m the spatial resolution requirement is assessed for the two
selected time-series:
hmin
hmin
(Resolution: 40 point)
(Resolution: 20 point)
18.2 sec
2.0 m
0.5 m
15.4 sec
2.8 m
0.7 m
Date
Tp
Time series 1: 19/04-06 16.15-16:45
Time series 2: 20/04-06 9.25-9:55
table 4
From table 4 it is observed that due to the long period nature of the dominant waves in the model,
minimum wavelength resolution requirements (20 points) is satisfied into the swash zone at a water
depth of 0.5-0.7 m. As the shallowest measuring station (AQDP 2) is positioned at a water depth of
7.4 m, it is assumed that slacking on the resolution requirement in the inner swash zone won’t have
a notable impact on the model predictions at any of the measuring stations.
The time step dt should be selected so that the minimum wave period occurring in the model should
be resolved by 25-35 time steps, while still at all locations satisfying the requirement given by the
Courant number:
Crmax  c max 
dt
1
dx
(33.)
where
c max 
g T
 tanh( khmax )
2 
The minimum wave period that can be resolved correctly at locations where instruments are
deployed, is at AQDP 2, where Tmin = 3.13 sec as governed by the Mike21 BW kh-limitation.
Selecting a dt of 0.1 sec will thus allow for a 31 step resolution of the minimum wave period while
keeping the maximum Courant number (at the offshore boundary) at less than 0.72.
47
Next to be decided is how large a geographical area to include in the computational domain.
The most common approach in MIKE21 BW is to generate waves is by internal generation along a
defined line inside the model. All open boundaries (such as open ocean) are closed using walls of
artificial land and the wave action in front of these walls are damped by placing a sponge layer in
front of each one. As a result no artificial reflection from the boundaries will be generated, which
in that case could seriously distort the wave action inside the model, but in return the waves close to
the boundary will dissipate and diffract into the sponge layer making the model predictions in the
close proximity non-physical.
As a result care must be taken to make the model domain large enough so that the area of interest
(Waimea Bay) is located far enough away from the boundaries in order not to experience any of
these undesirable effects. On the other hand the size of the model domain should be kept within
reasonable limits in order not to make computational run-times within realistic limits (max 30
hours).
Domain 1
Domain size: 700 x 941 = 658.700 grid point
In the first attempt to set up the computational domain it was decided that the domain should be
rotated 30 degrees making the shore-normal of the bay perpendicular to the wave generation line
located along the “left” side of the domain. The domain is set to have a height corresponding to
approximately 3 times the width of the Waimea Bay entry assuming that this height is sufficient to
avoid non-physical effects from the upper and lower boundary. The width of the domain is selected
so the landward boundary is wide enough (approximately 1 bay length) not to cause artificial
reflection from the right side back wall, as water close to the shoreline to some extent will
propagate into the beach-slope giving rise to unrealistic reflection if the landward boundary is too
close.
The left side boundary is set well offshore of the location of the ADCP allowing for a sufficiently
wide sponge layer an in attempt to avoid non-physical processes occurring close to the boundary
will have any harmful effects to the wave processes along the generation line.
In this model approach it has been emphasized that the wave generation line should run through the
location of the ADCP enforcing that the wave conditions at this distance from the bay are exactly,
what was originally measured by the instrument. As earlier mentioned the instrument is placed in
21 m of water so all water depths deeper than this is set to 21 m in model bathymetry.
48
figure 46
However from figure 46 it is noticed how the bathymetry in the top and (especially) in the bottom
of the domain is shallower than 21 m and must be altered in order to satisfy the requirement of
uniform water depth along the generation line. It is fully understood that such alteration of the
bathymetry could to some extend affect the accuracy of the model predictions inside Waimea Bay.
But as it was of interest to carry out the simulation with the ADCP location placed on the generation
line, this domain setup has been kept and thereby avoids eventual transformations in the wave train
while it passes flat bottom if the generation line was placed further offshore.
Domain 2.
Domain size: 1169 x 1201 = 1.403.969 grid point
As mentioned earlier the alteration in the bathymetry could pose distortions to model results. So
could the eventually too close proximity of the upper and lower boundary and the energy
49
dissipating effects (due to time extrapolation of the Boussinesq cross-terms) due to the angled wave
propagation.
As a result a second and much larger domain has been generated that due to its orientation does not
need a unidirectional input at the generation line and by enforcing a deepwater limit of 21 m, no
alteration of the near shore bathymetry is necessary. However the computational time for this model
is more than double than when using “Domain 1”. As a result Domain 2 has only been applied to
time series 2 (Thursday) as verification of the model results obtained using Domain 1.
Figure 47
50
4.2.2 Generating the bathymetry
The bathymetry for Waimea Bay was surveyed in year 2000 by airplane-mounted LIDAR and is
collected from the shoreline to 1 km offshore at 3-5 m spacing. Topographic data are collected from
the shoreline to 0.5 km inland at 1 m spacing. The depth precision is given to +/- 10 cm (JALBTCE
web page). A problem with the LIDAR data is that it cannot resolve the bathymetry in areas of high
turbidity such as the shore-break and wave induced turbidity around steep cliffs and rocky outcrops.
Secondly many of the small rocky outcrops along especially the south end of the bay is only
resolved partially and some does not even appear due to insufficient overlapping between the
bathymetry and topography surveys. As a result, great care had to be taken in order to perform a
correct interpolation of the unresolved “shadow” areas and also make sure that the rocky outcrops
are represented correctly in the model bathymetry.
figure 48 – Satisfactory representation
figure 49 – Insufficiently accurate representation
figure 48 and figure 49 represents two different attempts to resolve the two rocky outcrops at the
south entrance of the bay and poses an example of what problems were occurred when trying to
have features like these represented correctly in the model domain.
51
figure 50
figure 51
In this study it was not accomplished to setup an interpolation scheme in the MIKE Zero’s
imbedded Bathymetry editor that could resolve these features sufficiently correctly. Instead the
“nearest neighbor” interpolation method was used in the open sourced program “Generic Mapping
Tools” (GMT) to carry out the interpolation scheme and generate a new equidistant bathymetric
dataset that could be imported into the MIKE Bathymetry Editor. Subsequently these coastal
features appearing in the model were compared with photos and to some extend manually
smoothened and adjusted in order to obtain the most realistic representation.
Another concern was caused by the fact that while the bathymetry was surveyed in 2000, the
topography was surveyed in 2003. In addition, from a water depth of 0-3 meters in front of the
Waimea bay beach, no bathymetry data is available most likely due to high turbidity in this region
during the survey campaign in 2000 (see figure 55). The missing bathymetry points correspond to a
20-25 wide band that extend along the entire width of the bay.
52
figure 52
However as the bathymetry in this unresolved area is assumed to be fairly uniformly sloping, it was
decided to resolve this gap by linear interpolation.
But from visual inspection it was soon discovered that the data gap between the topography and
bathymetry data and the following interpolation resulted in a large rock in the south end of the bay
to be entirely excluded from the generated bathymetry (compare figure 53 and figure 54). However
even though the rock is notable in size its overall contribution to the wave transformations in the
bay must considered to be insignificant (see figure 55).
53
figure 53
figure 54
As a result no action has been taken just for the purpose of manually surveying and including this
feature in the model bathymetry.
figure 55
54
4.2.3 The sponge layer
The sponge layer has to be capable of absorbing even the most energetic wave from each of the two
time-series that is to be used in the Waimea bay model. As a general guideline good absorbing
characteristics has been obtained with a sponge layer L/2 wide, where L is the wavelength.
In order to make sure the sponge layer performs sufficiently, a flat-bottomed channel-simulation has
been carried out using a water depth of 21 m.
A fully reflecting wall has been placed in the right side of the channel and an absorbing sponge
layer has been placed behind the internal wave generation line in the other end of the channel.
The width of the sponge layer has been determined based on the spectral wave peak period of 18.2
sec from the Wednesday time-series corresponding to a wavelength of 250 m at h=21 m.
Hence the width of the sponge is set to 125 m (63 grid points) corresponding to the half-length. The
sponge layers is assessed for the two monochromatic input wave conditions based on the peak
period Tp and maximum wave height Hmax occurring in the two time series:
Wave ínput 1:
Wave input 2
T = 18.2 sec, H= 1.10 m
T=15.4 sec, H = 3.06 m
The length of the channel has been set to 1000 m and the grid spacing and time stepping are the
same as for the Waimea Bay model runs (dx =2 m, dt=0.01).
figure 56
figure 57
From figure 56 and figure 57 it is observed how the standing wave plots suggests a successfully
damping and dissipation of wave action at the offshore boundary by the sponge for both test cases
(no accumulation of wave energy).
55
However the asymmetrical shape of the envelope plots confirms the expectation of non-linearity as
discussed in section 4.1.5. In order to justify this assumption a new simulation was carried out with
same wave conditions as in figure 56 but with a water depth of 150 m. The time-step dt had to be
lowered to 0.05 sec in order to keep the Courant number below 1.
figure 58
In this case it is observed how linear conditions are obtained for the deep water case (Ur=0.12) and
that the sponge layer is still working effectively even though its thickness in this simulation only
corresponds to about 1/3 wavelength. The conclusion of this section is that a sponge layer with the
width of L/2 is sufficient for these two wave time-series that unfortunately are not entirely linear at
the model offshore boundary. Thus the width of the sponge layers used in these simulations will be
125 m (63 grid points) wide. In order to assure stabile model conditions inside the sponge a uniform
bathymetry over its cross-section is induced so that the last point outside the sponge is kept constant
along its cross-section (see figure 59):
56
figure 59
4.2.4 Further discussion of the non-linear boundary conditions.
figure 60
figure 61
In order to look more closely on the non-linear behavior of the input wave conditions at the offshore
boundary, a new channel simulation has been conducted using the same wave input as for the
sponge test, but with damping at both boundaries hence generating a free propagating wave on a flat
bottom. In figure 60 and figure 61 it is observed how a free second order harmonic and even a tiny
3rd order harmonic are released in the flume as a result of the introduction of a linear wave into the
non-linear environment. The small “curtain” appearing off each spectral peak and is caused by the
57
problem in using the FFT-method (Fast Fourier Transform) to resolve a finite time-series (Smith
1999).
figure 62
figure 63
From the time-domain plots in figure 62 and figure 63 it is clear how the higher order contributions
affect the propagating wave train. It can be seen that it is mainly the influence of the 3rd order
harmonic that causes the spatially varying wave height in the channel.
figure 64
When plotting the spatial wave height distribution over the flat-bottomed channel it can be observed
how the wave-height is fluctuating with as much as 8%. It should be recalled that these test cases
corresponds to the most extreme occurring incident waves occurring Wednesday and Thursday and
as the non-linearities decreases for smaller wave height it is expected that the overall wave height
inaccuracies in the Waimea Bay simulation due to this phenomenon should be expected to be
significantly smaller than observed in figure 64.
58
4.2.5 Filter layer
As numerical instabilities will occur in the swash zone in areas where the surface roller cannot be
resolved, a filter layer is applied to the Waimea Bay model. Due to the complex distribution of
wave breaking occurring in the model, the filter layer is applied from a water depth of 3 meters at
the steep cliffs and 1 m and the beach slope of Waimea bay. It is acknowledged that in areas where
the filter had to be extended out to 3 m, artificially induced wave damping will occur over a
significant area affecting the reliability of the model predictions in these areas. It can only be hoped
that this would only insignificantly affect the overall model predictions inside the bay.
From previous good experience in the preliminary study a filter coefficient of 0.5 is used. Inside the
bay from a water depth of 1.0-1.5 m, the filter coefficient is undergoing a smooth transition to 0
using linear interpolation.
figure 65
4.2.6 Porosity layer
Porosity layers are usually applied to simulate partial reflection from rubble mound breakwaters but
are in this study in some simulations applied to simulate partial reflection from the steep cliffs in the
domain. The porosity layer is applied by substituting the “porous structure” with a porosity layer
setting the water depth at this location to equal the water depth in front of the structure. A more
detailed description is given where Porosity is applied.
59
4.2.7 Additional parameters
Due to the large number of adjustment options provided in MIKE21 BW, the huge model run-times
and the limited opportunity to assess the value of every single one, a number of parameters have
been kept unchanged throughout this study. This concerns the moving shoreline and wave breaking
parameters. This decision to keep these unchanged is based on the experiences in the preliminary
study that suggests that an alteration of these values would not notably change model results.
The parameters are:
Moving shore
Slot depth: -21
Slot width: 0.2
Slot smoothing parameter: 100
Slot friction coefficient: 0.2
Wave breaking
Roller form factor: 2 (plunging breakers)
Type of roller celerity: Type 3
Roller celerity factor: 1.3
Initial breaking angle: 20
Final breaking angle: 10
Half-time for cut-off roller: 3
Additionally the time extrapolation factor is kept at 0.8, which is the heights possible value assuring
a stable model simulation.
The last model parameter to be mentioned is the bottom friction factor defined by the Chezy
number C. As it is not possible to determine an accurately uniform value for Waimea Bay, a
number of different values have been tested in the simulations in order to obtain the most correct
representative value.
60
4.3
Simulation Results:
4.3.1 The first attempt.
Simulation info: Simulation 1
Data files: v25
Wave-input: Time-series 2
Incident wave direction: 327
Bottom friction: C=32
Bathymetry: Domain 1
Porosity map: none
For the first model run the times-series from Thursday morning is used as offshore boundary
condition. Being the most energetic wave input it was considered suitable for the initial simulation,
as it would set the upper bar for eventual model instabilities (Blow-ups) that could occur during
simulation.
figure 66
Hm0
Field Data Mike21 BW Diff. [%]
Hm0
1.93 m
2.18 m
13 %
Offshore
1.60 m
19 %
Bay Entry 1.35 m
2.06 m
30 %
Bay Middle 1.59 m
table 5
61
Additionally, the larger waves in this time-series will induce a much more pronounced surf zone in
the domain that to some extend can be compared in time and space to recorded video from within
the same timeframe. Also during this energetic time-series, processes such as wave reflection from
the landward boundaries are much easier visually detectable from the model results.
In figure 66 the surface elevation time-series from three field stations are decomposed into their
harmonic components and presented in frequency space where they are compared in to model
results at these exact locations. Based on these spectra, Hm0 has been calculated and compared.
The zero-upcrossing method has been applied to extract Hm and Hsig from the surface elevation
time-series. Also the mean, minimum and maximum velocities have been extracted from the Uand V-velocity time-series. As Mike21 BW outputs the depth-integrated values P and Q, these have
been divided with their corresponding water depths (e.g. U (t) =P (t)/(h+(t)) to provide the correct
representation for comparison with the field measurements given in depth averaged values. The U
and V time-series from the field measurements have been rotated 30 degrees (clock-wise) to match
the axis orientation in the domain.
Mean Wave height
Field Data
Hm
1.22 m
Offshore
Bay Entry 0.84 m
Bay Middle 0.94 m
Mike21 BW
1.38 m
0.99 m
1.29 m
Diff. [%]
13 %
17.1 %
27.1 %
Sig. Wave height
Field Data
Hsig
1.84 m
Offshore
Bay Entry 1.30 m
Bay Middle 1.48 m
table 6
Velocities offshore
Umax
Field Data
1.07 m/s
Mike21 BW 0.92 m/s
Mike21 BW
2.05 m
1.55 m
2.02 m
Diff. [%]
11 %
19 %
36 %
table 7
Umin
Umean
Vmax
Vmin
Vmean
-0.87 m/s -0.01 m/s 0.50 m/s 0.69 m/s 0.00 m/s
-1.07 m/s -0.05 m/s -1.10 m/s -0.38 m/s 0.01 m/s
table 8
The offshore location
In figure 66 it is observed that the spectral peak at the location of the ADCP (offshore) is clearly
overestimated by the model. The result is an overestimation of the corresponding Hm0. Also an
offshore overestimation of Hm and Hsig (presented in table 6and table 7) is clearly noticed.
The most likely explanation for this is that the overestimated model wave height is caused by wave
reflection. Especially from the steep cliffs and rocky outcrops around the entry of the bay.
62
figure 67
In figure 67 it is observed how extensive wave reflection occurs from the cliffs of the northern bay
entry and the two rocky outcrops outside the south end of the bay entry producing scattered waves
that unquestionably affects the model output in both the offshore station and in the bay entry.
The first question that arises is if the models prediction of the magnitude of this reflection is correct.
This also yields the question if the input surface elevation time-series measured by the ADCP in
fact can be treated as a purely incident wave input or if the signal also contains a significant
contribution from reflected waves traveling in an adverse direction (not necessarily the direct
opposite direction) of the incident wave train. If significant wave reflection indeed does occur
during the field campaign, it is not straightforward to tell if the surface elevation data collected in a
63
particular point is magnified or inhibited by this phenomenon. It entirely depends if the point is
close to a node or half-node in the system (e.g. recall figure 58).
In order to look deeper into this matter, a comparison between the measured and calculated depthintegrated velocities must be taken into consideration.
figure 68
From comparison of the offshore velocity scatter plots in figure 68 there seems to be a good
agreement between filed measurement and model predictions and no apparent cross-directional
velocity components are visible suggesting that the most dominant contribution to the signal from
reflected waves occurs in the adverse direction of the wave direction. If this assumption is to be
related to figure 67 it would be suggested that the dominant reflection registered in Station 1 mainly
occurs from the two rocky outcrops outside south end of the bay as well as from the cliffs along the
south entry of the bay itself.
64
figure 69
figure 70
By performing a time-series comparison in Station 1, the influence of reflection at this location is
quite evident. In figure 69 it is observed how the first few waves are well predicted.
But in figure 70 it is observed how deviations stats to occur as the input time series shows
indications of wave reflection as small “bumps” in the wave troughs and small sudden “drops” in
the wave crests, which is not in the same manner represented in the model time-series. This initially
suggests that the model at this location does not predict wave reflection that is actually occurring in
the bay. However from figure 67 it is confirmed that the model does predict reflected waves to be
propagating in the direction of Station 1 so it could also just be that it is the phase shift between the
reflected and incident wave that is not correctly predicted. From figure 70 it could be suggested that
the model predicts the reflected wave train to be in phase with the incident wave train resulting in
wave height amplification instead of a reduction.
However the only thing that can be concluded at this point is that significant reflection occurs in
both the model domain and apparently also in the physical domain and that the wave signal in
Station 1 in the model is sensitive to this reflection. It will be left to the following model runs to
attempt to resolve this question in further detail.
The bay Entry
From figure 66 it is observed how the spectral peak is well predicted by the model at the bay entry
location (STATION 2). It is also noticed how the wave conditions at this location has become more
non-linear due to the shallower water depth, as a more pronounced 2nd order harmonic amplitude is
65
now clearly visible in the spectrum. However it is also noticed that the model slightly overestimates
the magnitude of the 2nd order harmonic. The overall result is a small overestimation in Hm0, Hm and
Hsig.
Velocities Bay Entry
Umax
Field Data
1.05 m/s
Mike21 BW 1.02 m/s
Umin
Umean
Vmax
-0.88 m/s -0.02 m/s 0.33 m/s
-1.10 m/s -0.01 m/s 0.46 m/s
Vmin
Vmean
-0.33 m/s 0.01 m/s
-0.44 m/s 0.01 m/s
table 9
When comparing the velocity amplitudes presented in table 9 it is observed how a good agreement
exists between model results and field data for Umin and Umax while the model overestimates Vmin
and Vmax with a factor 2. This suggests that the model overestimates the amount of wave reflection
occurring at the southern entrance of the bay (recall figure 67), where the incident waves “bounces
off” the northern and southern cliff sending a reflected wave traveling along the cross-section of the
bay entrance*.
figure 71
In the velocity scatter plot from the bay entry location (figure 71), the directional agreement
between model and field data is general in good agreement. It is noticed how the waves have now
refracted almost 30 degrees since leaving the wave generation line and are now traveling almost inline with the shore normal. However in the model predictions occasional minor deviations from this
narrow band direction can be noticed, which most likely are caused by the previously mentioned
reflected waves traveling cross-directional of the primary wave direction. It is noticed that this
phenomenon does not show up in the field data.
*
Due to the bays orientation in the model, U is parallel to the shore normal.
66
figure 73
figure 72
In figure 72 it is illustrated how the first 3 waves (the initial forerunner excluded) that reach station
2 in the bay entry are in very good agreement with field measurements. But subsequently the
agreement starts to deteriorate. While the agreement between the two time-series always remains
fair, it is clearly visible that something is distorting the model predicted signal sometimes causing
an overestimated crest or through (or vice versa) and occasionally even producing a wave that does
not seem to be in any coherence with field measurements.
The fact that the time-series agreement is good for about 40 seconds after the first wave passes
station 2 and then deteriorates strongly suggests that the way in which the waves are subsequently
being dissipated and/or reflected at the landward boundaries are not represented correctly with the
initial model setup.
Velocities Bay Middle
Umax
Umin
Umean
Vmax
Field Data
1.35 m/s -1.26 m/s -0.03 m/s 0.46 m/s
Mike21 BW 1.40 m/s -1.58 m/s 0.01 m/s 1.56 m/s
Vmin
Vmean
-0.58 m/s -0.06 m/s
-0.91 m/s -0.01 m/s
table 10
The Bay Middle
In figure 66 it is apparent how the non-linearity of the wave signal at this location is even stronger
(compared to the bay entry) as now also small 3rd order harmonic amplitudes has become visible in
the wave spectra. Unfortunately the model significantly overestimates the higher order harmonics
resulting in overestimating of Hm0, Hm and Hsig of 28-36%. But from the comparison of velocity
amplitudes in table 10 it is again clear that while Umax and Umin are in order of magnitude with field
measurements, Vmax and Vmin are overestimated with as much as a factor 2-3.
67
Again these results strongly suggests that the model clearly overestimates the wave reflection
occurring at the southern end of the bay, which results in the clearly defined waves traveling along
the bay cross-section.
figure 74
From figure 74 it is observed how there exist a strong mismatch between field data and model
results, and it is very clear that the significant cross-sectional wave motion occurring in the model
(generating extensive directional spreading in the scatter plot) is not physically represented in the
field data, where the direction is very narrow banded and almost inline with the shore normal.
figure 75
figure 76
68
The cross-sectional wave propagation is illustrated in figure 75, where a notable fraction of the
incident wave energy is reflected from the southern cliff walls of the bay and right into the direction
of the two instruments at station 2 and 3.
figure 77
figure 78
Finally from the time-series comparison in figure 77 and figure 78 it is observed, how the coherence
is heavily distorted by this phenomenon and frequently no apparent coherence seems to exist at all.
figure 79
From the spatial plot of Hm0 in figure 76 it is seen how a good indication of the presence of wave
reflection is apparent, being visually distinguishable as the strongly oscillating wave height patterns
running perpendicular to the land boundary at a number of areas in the domain. As expected a
pronounced standing wave pattern is located off the south side of the bay, but also a similar pattern
69
is noticed running parallel to shore normal of the Waimea beach indicating significant reflection
also occurring from the beach profile itself. In figure 79 a shore normal standing wave pattern
inside Waimea Bay is clearly distinguishable. Again at this stage it is hard from only two
measurement stations inside the bay to determine if this standing wave pattern is in fact occurring
during the field campaign. The only thing that can be concluded at this stage is that the crosssectional wave motion in the bay is overestimated with the current model setup (recall figure 71 and
figure 74).
Remarks to simulation 1:
The total model run is 18000 time-steps However with the current setting the simulation blows up
after 14300 time steps due to extraordinary cross-sectional wave action inside the bay. After lots of
unsuccessful attempts to make the model run stable for its full duration, a stabile setting was found
by decreasing both the Time-extrapolation factor to 0.5, the roller form factor to 1.5 and extending
the sponge layer inside the bay to a water depth of 3 m (instead of just 1 m). However these changes
only contribute to a damped solution of the former and the resulting simulation is considered a
much poorer representation of the actual physical processes occurring in the domain. As a result it
was considered more correct to compare the first 14300 time steps of the original simulation with
the corresponding field results and from there clarify the problems that spawned the following
attempt to improve model results.
70
4.3.2 Introducing porosity layer
Simulation info: simulation 2
Data files: v28
Wave-input: Time-series 2
Incident wave direction: 327
Bottom friction: C=35
Bathymetry: Domain 1
Porosity map: porosity map 1 (cliffs + rocky outcrops)
In this simulation a porosity layer has been applied along all the steep cliffs in the domain and the
rocky outcrops along the southern entry of the bay. By introducing porosity layers along these parts
of the landward boundary a reduced reflection at these locations is enforced, in an attempt to
improve model predictions.
figure 81
figure 80
The porosity layers were designed as suggested in (DHI Software 2004) being 6-8 grid points wide
(12-16 m) corresponding to approximately L/4 with wave period Tp in a water depth of 1 m which
in this setup was considered the “toe” of the structure The reflection coefficient was set to 0.8
71
assuming a “smooth impermeable slope”. An illustration of the selected porosity layer setup is
given in figure 81.
figure 82
Hm0
Hm0
Offshore
Bay Entry
Bay Middle
Field Data
1.85 m
1.39 m
1.68 m
Mike21 BW
2.03 m
1.74 m
1.68 m
Diff. [%]
9.7 %
25 %
0%
figure 83
From figure 82 it is observed how the implementation of the porosity layer provides a desired
decrease in the spectral peak offshore and also significantly improves the prediction of the higher
order harmonics in the bay middle. However on the contrary it is now observed that the amplitude
of the spectral peak frequency in the bay middle (correctly represented in simulation 1) has
decreased. At the same time the spectral peak and its 2nd order harmonic in the bay entry have been
increased compared to simulation 1.
72
Mean Wave height
Field Data
Hm
1.15 m
Offshore
Bay Entry 0.89 m
Bay Middle 1.00 m
Mike21 BW
1.30 m
1.10 m
1.07 m
Diff. [%]
13 %
24 %
7%
table 11
Sig. Wave height
Field Data
Hsig
1.80 m
Offshore
Bay Entry 1.37 m
Bay Middle 1.61 m
Mike21 BW
1.95 m
1.73 m
1.72 m
Diff. [%]
8.3 %
26%
7%
table 12
Again in table 11 and table 12 (based on the zero-upcrossing time-series analysis) it is observed
how prediction of the wave conditions offshore and in the bay middle has been significantly
improved by applying the porosity layer. However again it is noticed that the predicted wave height
in the bay entry has increased compared to Simulation 1, where no porosity layer was applied.
figure 84
From the Hm0 plot in figure 84 it is observed how the assumed standing wave patterns are far from
as apparent as for simulation 1 (figure 76) and thus the porosity map seems successful in damping
the reflection from the steep cliffs and rocky outcrops. However, clearly notable standing wave
patterns are still noticed from especially the south corner of the bay rocky outcrops but also along
the beach in the bottom of Waimea bay.
73
figure 85
figure 86
When plotting the in-line wave height distribution along the shore-normal it is observed how the
implementation of the porosity map to some extent have damped the standing wave pattern inside
the bay compared to simulation 1. However it is observed how especially Station 3 is located in an
area with very large wave height gradients making it evident that a correct prediction of the
magnitude and 2D-distribution of wave reflection occurring in the bay is very important to the
accuracy of model predictions at this location.
Velocities offshore
Umax
Field Data
1.07 m/s
Mike21 BW 0.81 m/s
Umin
Umean
Vmax
Vmin
Vmean
-0.87 m/s -0.01 m/s 0.50 m/s -0.62 m/s 0.00 m/s
-1.12 m/s -0.03 m/s -0.59 m/s -0.44 m/s 0.01 m/s
table 13
Velocities Bay Entry
Umax
Field Data
1.05 m/s
Mike21 BW 1.22 m/s
Umin
Umean
Vmax
-0.88 m/s -0.02 m/s 0.33 m/s
-1.10 m/s -0.01 m/s 0.51 m/s
Vmin
Vmean
-0.33 m/s 0.01 m/s
-0.52 m/s 0 m/s
table 14
Velocities Bay Middle
Umax
Umin
Umean
Vmax
Vmin
Vmean
Field Data
1.35 m/s -1.26 m/s -0.03 m/s 0.46 m/s -0.58 m/s -0.06 m/s
Mike21 BW 1.64 m/s -1.78 m/s -0.01 m/s 1.27 m/s -0.80 m/s -0.01 m/s
table 15
From table 13, table 14 and table 15 it is noticed that the mean and peak velocity values are rather
unchanged compared to simulation 1 with the exception of the cross-sectional peak velocities that
have decreased significantly compared to simulation 1 and are now in much better agreement with
field data.
74
figure 87
figure 89
figure 88
When considering the scatter plots from these two locations it is obvious that introducing a porosity
layer in the model domain has made a significant improvement in resolving the cross-sectional
reflection in the bay middle as observed in figure 89. However some cross-sectional oscillation is
still noticed from waves reflected from the southern end of the bay and it is possible that this could
be eliminated if a more absorbing porosity layer were applied in along the south side of the bay that
appears to be the source of this reflection affecting the velocity prediction in Station 3.
figure 90
figure 91
When comparing the times-series from Simulation 1 and 2 at the offshore location a slightly better
agreement has been obtained by introducing a porosity map. However major features in the field
data time series (such as assumable reflected waves appearing as “trough bumps” in figure 90) is
still not well represented by the model predictions. But what is very interesting is that the locations
where the “trough bumps” are not predicted, the adjacent wave crests are usually overestimated
with the same order of magnitude. This could suggest that the lack of coherence is caused by a
small phase mismatch between the incident wave and the predicted reflected wave.
75
figure 92
figure 93
At station 2 in the bay entry it is noticed that in spite of the implementation of the porosity map
there is still occasional difficulties with representing the reflection at this point correctly.
figure 94
figure 95
Also at station 3 in the Bay middle it is clear how the implementation of the porosity layer has lead
to improved wave height prediction at some locations (in the time series) and inaccurate predictions
elsewhere. But in spite of not being convincingly clear from figure 94 and figure 95, a much better
integral wave prediction is obtained in the bay middle by applying the porosity map. However it is
concluded that the difficulties with resolving the bay reflection correctly might not be as easily
solved by simply just introducing porosity map.
76
4.3.3 Extracting the reflected wave
Simulation info: Simulation 3
Data files: v31
Wave-input: Time-series 2 (extracted incident wave)
Incident wave direction: 327
Bottom friction: C=32
Bathymetry: Domain 1
Porosity map: porosity map 1
As mentioned previously it is suspected that the wave signal given by the ADCP and used as input
wave condition might contain a significant contribution from reflected waves off the landward
boundary. In an attempt to account for this matter a simple method by (Kubota et al. 1989) has been
applied on “time series 2” to extract the incident wave from the total surface elevation signal using
the velocity time series U0 and V0 from this location. The principle is based on the 1D-assumption
that the total wave signal can be considered a superposition of an incident and reflected wave train
traveling in the direct opposite direction of each other. Hence:
  I R
(34.)
u  uI  uR
(35.)
Due to the 1D-formulation u is here the projection of U0 and V0 on the primary wave direction.
Hence:
u  U 0  sin( 360    )  V 0  cos(360    )
(36.)
Further more it is assumed that the surface elevation can be sufficiently represented by a finite
number of free harmonics, thus:
N
 I   Ai cos i t   i 
(37.)
 R   Bi cos i t   i* 
(38.)
i 1
N
i 1
From these expressions the corresponding u-velocities are calculated from linear wave theory:
N
u I   H i Ai cos i t   i 
(39.)
i 1
N

u R   H i Bi cos  i t   i*

(40.)
i 1
Where
Hi 
 i cosh k i (h  z)
sinh k i h
(41.)
77
Using Fourier analysis the following expression is obtained for u:
N
u   C i cos  i t  Di sin  i t 
(42.)
i 1
Where Ci and Di are the Fourier coefficients.
Inserting (29), (30) and (32) into (25) you get:
N
N
i 1
i 1


N
 H i Ai cos i t   i    H i Bi cos i t   i*   Ci cos i t  Di sin i t 
(43.)
i 1
Recalling that each wave can be treated as free, both sides are divided through with Hi and by
inserting (27) and (28) you obtain:
N
 I  R  
i 1
1
Ci cos  i t  Di sin  i t 
Hi
(44.)
Finally by applying (24) you obtain:
I 
N

1 
1
   
 C i  cos( i  t )  Di  cos( i  t ) 
2 
i 1 H i

(45.)
From this equation the incident wave train can be extracted from the ADCP wave signal.
figure 96
By comparing the two time-series it does appear that this method is quite successful in filtering out
suspected contributions from a reflected wave train observed in “A” and “B”.
78
figure 97
Hm0
Hm0
Offshore
Bay Entry
Bay Middle
Field Data
1.85 m
1.39 m
1.68 m
Mike21 BW
2.00 m
1.70 m
1.60 m
Diff. [%]
8%
22 %
5%
table 16
Mean Wave height
Field Data
Hm
1.15 m
Offshore
Bay Entry 0.89 m
Bay Middle 1.00 m
Mike21 BW
1.25 m
1.10 m
1.01 m
Diff. [%]
9%
23%
1%
Sig. Wave height
Field Data
Hsig
1.80 m
Offshore
Bay Entry 1.37 m
Bay Middle 1.61 m
table 17
Mike21 BW
1.96 m
1.71 m
1.66 m
Diff. [%]
9%
20%
3%
table 18
It is noticed that using the produced “incident” time-series as offshore wave input lead to small
improvement in the prediction of Hm0, Hm and Hsig.
However the spectral peak at station 3 is still not correctly represented and the 2nd order harmonics
for station 2 and 3 are still overestimated. Also it does not solve the problem of what causes the
model predictions in station 2 to be clearly exaggerated.
79
Velocities offshore
Umax
Field Data
1.07 m/s
Mike21 BW 0.74 m/s
Umin
Umean
Vmax
-0.87 m/s -0.01 m/s 0.50 m/s
-0.90 m/s -0.03 m/s 0.47 m/s
Vmin
Vmean
-0.62 m/s 0.00 m/s
-0.46 m/s 0.01 m/s
table 19
Velocities Bay Entry
Umax
Field Data
1.05 m/s
Mike21 BW 1.00 m/s
Umin
Umean
Vmax
-0.88 m/s -0.02 m/s 0.33 m/s
-0.00
0.00 m/s 0.42 m/s
Vmin
Vmean
-0.33 m/s 0.01 m/s
-0.49 m/s 0.00 m/s
table 20
Velocities Bay Middle
Umax
Umin
Umean
Vmax
Vmin
Vmean
Field Data
1.35 m/s -1.26 m/s -0.03 m/s 0.46 m/s -0.58 m/s -0.06 m/s
Mike21 BW 1.25 m/s -1.40 m/s 0.02 m/s 1.19 m/s -0.62 m/s 0.00 m/s
table 21
From the tables above a small improvement of the velocity magnitudes at Station 2 and 3 is
obtained, while the model performance for the offshore prediction is rather unchanged.
figure 98
figure 99
figure 100
In the velocity scatter plots the same trend is evident as a slightly improved agreement with field
data.
80
figure 101
figure 102
figure 103
figure 105
figure 104
figure 106
81
For all the time series plots for Simulation 3 it can be observed that the influence of extracting the
incident wave is only small. Most significantly is the fact that the resultant incident wave is less
energetic in terms of wave amplitude compared to the “raw input” and as a result it leads to a
general decrease in predicted wave height at all 3 field stations.
figure 107
figure 108
However from a comparison of Hm0-plots it is observed that changing the wave input condition only
has a small influence on the wave transformation processes inside the bay.
The conclusion of this simulation is that even though providing slightly better model estimates,
extracting (and removing) the reflected wave train from the input wave conditions is not crucial in
solving the problems with resolving the wave reflection occurring inside the bay.
82
4.3.4 Increasing bottom friction
Simulation info: simulation 4
Data files: v33
Wave-input: Time-series 2
Incident wave direction: 327
Bottom friction: C=25
Bathymetry: Domain 1
Porosity map: none
This simulation is identical to “simulation 2” with the exception that the bottom friction in the
model has been increased (lower Chezy nr.). While it hardly ever was assumed that changing the
bottom friction alone could lead to significantly improved results, it was found necessary to
investigate and quantify the effect of a change in bottom friction on how it affected the models
predictions of velocities and surface elevation in the domain.
figure 109
Hm0
Hm0
Offshore
Bay Entry
Bay Middle
Field Data
1.85 m
1.40 m
1.68 m
Mike21 BW
2.03 m
1.72 m
1.65 m
Diff. [%]
10 %
23 %
2%
table 22
83
Mean Wave height
Field Data
Hm
1.15 m
Offshore
Bay Entry 0.89 m
Bay Middle 1.00 m
Mike21 BW
1.29 m
1.10 m
1.05 m
Diff. [%]
12%
24%
5%
Sig. Wave height
Field Data
Hsig
1.80 m
Offshore
Bay Entry 1.37 m
Bay Middle 1.61 m
table 23
Mike21 BW
1.94 m
1.72 m
1.69 m
Diff. [%]
8%
26%
5%
table 24
From the spectral plots in figure 109 it is observed that they are almost identical to those in
simulation 1 (figure 66) only insignificantly smaller in magnitude due to the increased energy
dissipating due to the increased bottom roughness. As a result Hm0, Hm and Hsig are slightly smaller
than in Simulation 1, which leads to a slight improvement of model results.
Velocities offshore
Umax
Field Data
1.07 m/s
Mike21 BW 0.82 m/s
Umin
Umean
Vmax
-0.87 m/s -0.01 m/s 0.50 m/s
-1.12 m/s -0.03 m/s 0.59 m/s
Vmin
Vmean
-0.62 m/s 0.00 m/s
-0.42 m/s 0.01 m/s
table 25
Velocities Bay Entry
Umax
Field Data
1.05 m/s
Mike21 BW 1.12 m/s
Umin
Umean
Vmax
-0.88 m/s -0.02 m/s 0.33 m/s
-0.96 m/s -0.01 m/s 0.46 m/s
Vmin
Vmean
-0.33 m/s 0.01 m/s
-0.47 m/s 0.00 m/s
table 26
Velocities Bay Middle
Umax
Umin
Umean
Vmax
Vmin
Vmean
Field Data
1.35 m/s -1.26 m/s -0.03 m/s 0.46 m/s -0.58 m/s -0.06 m/s
Mike21 BW 1.43 m/s -1.55 m/s 0.01 m/s 1.14 m/s -0.70 m/s 0.00 m/s
table 27
It is also noticed that the velocity amplitudes are slightly smaller in magnitude compared to
Simulation 2. However it is clear that an increase in bottom friction does not help in improving the
inaccurate model prediction in Station 2.
84
figure 110
figure 111
figure 112
Again the velocity scatter plots confirm the sight improvement in velocity prediction for Station 3
in the bay middle and to a lesser degree also for Station 2 in the bay entry. The offshore velocity
prediction remains rather unaffected.
figure 113
figure 114
figure 115
However, when examining the time-series plots it is evident that changing the bottom friction
virtually has no effect on the models wave height prediction for the selected station locations. The
most obvious explanation for this is that the wave driven current values are very small and almost
insignificant, thus not affecting the wave height at these locations as e.g. was the case in the rip
current simulation in the preliminary study.
85
Also from a comparison of Hm0 plots it is observed that changing the bottom friction has almost no
effect on the wave transformation processes occurring in the bay.
86
4.3.5 Enlarging the domain
Simulation info: simulation 5
Data files: newbat 6
Wave-input: Time-series 2
Incident wave direction: 0  (327)
Bottom friction: C=35
Bathymetry: Domain 2
Porosity map: porosity map 1
figure 116
Hm0
Hm0
Offshore
Bay Entry
Bay Middle
Field Data
1.85 m
1.40 m
1.68 m
Mike21 BW
1.90 m
1.78 m
1.81 m
Diff. [%]
3%
27%
8%
table 28
Mean Wave height
Field Data
Hm
1.15 m
Offshore
Bay Entry 0.89 m
Bay Middle 1.00 m
Mike21 BW
1.18 m
1.10 m
1.11 m
table 29
Diff. [%]
3%
24%
11%
Sig. Wave height
Field Data
Hsig
1.80 m
Offshore
Bay Entry 1.37 m
Bay Middle 1.61 m
Mike21 BW
1.83 m
1.75 m
1.78 m
Diff. [%]
2%
28%
11%
table 30
87
From the spectral plots and the wave tables it is noticed that the wave height predictions at Station 1
has improved and is now in very good agreement with field data. But at the same time it is noticed
that the larger domain and the domain-perpendicular wave propagation does not improve results
inside the bay where the wave height in Station 2 is still significantly overestimated the spectral
peak in Station 3 is still underestimated.
figure 117
From the Hm0-plot in figure 117 it is observed how the accurate wave height prediction in Station 1
might me a coincidence as the significant wave height offshore fluctuates between
1.83 m and
2.15 m in spite of the uniform conditions and flat bottom. From the Hm0-plot it looks like the
spatial variation in wave height is greatly influenced by extensive reflection from especially the
southern end of Waimea bay and the two large rocky outcrops in front of the southern entrance. But
also from figure 64 it should be recalled that due to the weakly non-linearity of “time-series 2”
(used as wave-input) the interaction between the (assumed free) primary and 2nd order harmonic
will alone generate a spatially fluctuating wave height of up to 8%.
88
figure 118 – Bay Close-up, Simulation 2
figure 119 – Bay Close-up, Simulation 5
From Hm0 comparison with simulation 2, it is observed that no significant change in predicted
wave height has occurred by applying Domain 2. The small deviations between the two plots are
considered to be mostly due to small inconsistencies in the applied porosity layers that each had to
be applied manually
Velocities offshore
Umax
Field Data
1.07 m/s
Mike21 BW 0.86 m/s
Umin
Umean
Vmax
-0.87 m/s -0.01 m/s 0.50 m/s
-0.92 m/s 0.00 m/s 0.39 m/s
Vmin
Vmean
-0.62 m/s 0.00 m/s
-0.39 m/s 0.00 m/s
table 31
Velocities Bay Entry
Umax
Field Data
1.05 m/s
Mike21 BW 1.01 m/s
Umin
Umean
Vmax
-0.88 m/s -0.02 m/s 0.33 m/s
-0.90 m/s 0.00 m/s 0.59 m/s
Vmin
Vmean
-0.33 m/s 0.01 m/s
-0.56 m/s -0.01 m/s
table 32
Velocities Bay Middle
Umax
Umin
Umean
Vmax
Vmin
Vmean
Field Data
1.35 m/s -1.26 m/s -0.03 m/s 0.46 m/s -0.58 m/s -0.06 m/s
Mike21 BW 1.17 m/s -1.44 m/s 0.01 m/s 0.78 m/s -1.02 m/s -0.02 m/s
table 33
89
figure 120
figure 122
figure 121
As no significant difference in the wave field is noticed it is not a surprise that the velocity
magnitudes and the directional agreement are unchanged for this simulation.
figure 123
figure 124
In the time-series comparison for Station 1 it is observed how the wave amplitudes are in slightly
better agreement using domain 2, but still it is noticed that features such as occasional “bumps” in
the crest and trough are predicted more consistently in this simulation.
90
Figure 125
Figure 126
At Station 2 at the bay entry it is again observed how the time-series from Simulation 2 and 5 are
almost identical. However at some locations, small variations are noticed mostly just as a variation
in wave amplitude but these variations are mainly considered to be due to small inconsistencies
between the two porosity maps.
Figure 127
Figure 128
At Station 3 in the bay middle the same trend is observed as for Station 2.
Based on Simulation 5 it has been observed that even though a better wave height estimation is
obtained using Domain 2, it does not in any significant way change the models prediction of the
wave height distribution inside the bay, nor does it improve the overall correlation with field data
(in Station 2 and 3). It is acknowledged that running the model on domain 2 has the potential of
assuring more reliable results because the area of interest (the bay) is further away from the
91
boundaries and also because of the fact that no artificial damping of the Boussinesq cross-terms will
be experienced since the dominant wave propagation occur perpendicular to the domain. However
on the other hand a 30-minute simulation using Domain 2 takes 35 hours to complete compared to
“only” 11 hours using domain 1.
And as the two model predictions provides virtually the same predictions it has been decided that
Domain 1 is a sufficiently reliable platform for the model simulations in this study.
92
4.3.6 Altering the wave direction
Simulation info: simulation 6
Data files: v34
Wave-input: Time-series 2
Incident wave direction: 330
Bottom friction: C=32
Bathymetry: Domain 1
Porosity map: porosity map 1
As mentioned previously the dominant wave direction at the offshore boundary is determined from
the velocity measurements gathered in Station 1 by the ADCP. The instrument generates its
directional output using its imbedded compass. However, it was informed by laboratory staff that
this compass could only be expected an accuracy margin within +/- 2-3 degrees. As a result a
simulation with an incident wave direction of 330 in order to assess the models response sensitivity
to this change.
figure 129
93
Hm0
Hm0
Offshore
Bay Entry
Bay Middle
Field Data
1.85 m
1.40 m
1.68 m
Mike21 BW
1.99 m
1.60 m
1.62 m
Diff. [%]
8%
14%
4%
table 34
Mean Wave height
Field Data
Hm
1.15 m
Offshore
Bay Entry 0.89 m
Bay Middle 1.00 m
Mike21 BW
1.27 m
1.00 m
1.07 m
Diff. [%]
10%
12%
7%
Sig. Wave height
Field Data
Hsig
1.80 m
Offshore
Bay Entry 1.37 m
Bay Middle 1.61 m
table 35
Mike21 BW
1.94 m
1.59 m
1.68 m
Diff. [%]
8%
16%
4%
table 36
The most noticeable observation from the wave tables and spectral plots is that the change in
incident wave angle significantly increased the accuracy of the models wave prediction in Station 2,
while the agreement in Station 1 and 3 remains almost unchanged.
figure 130
figure 131
The explanation for this is found by comparing the Hm0 plot with the one from Simulation 2. As
recalled the only difference between these two simulations is an incident wave angle of 3. As
experienced in all previous simulations, Station 2 seems to be located on the ridge of a sheltered
area (a “shadow zone”) that on average does not experience wave heights as large the rest of the
respective cross-section in the bay.
94
figure 132
As noticed in figure 132 the incident waves will experience significant refraction immediate
offshore of the north end of the bay due to the presence of the reef, where the waves will wrap
around, shoal and break leaving the immediate area of Station 2 much more sheltered. The south
entrance of the bay lies fair unprotected at this wave direction and the waves arriving here will not
have undergone much refraction since their generation at the offshore boundary. The result is a
small sheltered zone at the bay entrance just leeward of the reef where the wave heights in general
are smaller. And as Station 2 apparently is located in the rim of this zone, even a small change in
incident wave direction (3) can have a tremendous effect on the predicted wave heights in this
point. During times of small to medium sized swell action (such as “time-series 2” Thursday
morning) the reef also known as “Pinball's” provides a popular spot for surfing. During these
occasions the zone just leeward of the reef does provide a “safe” passage for surfers when paddling
out into the surf spot located at the bay entry, as the waves in this zone are usually smaller and are
not breaking. So the models prediction of this sheltered zone does seem realistic estimated from
visual observations.
95
Waimea bay south end – Thursday AM, April 20. 2006
Waimea bay north end – Thursday AM, April 20. 2006
However due to the severely limited amount of instruments inside the bay (2!) it is not possible in
this study scientifically to document and verify the existence of this “sheltered” zone as predicted
by the model.
Velocities offshore
Umax
Field Data
1.07 m/s
Mike21 BW 0.88 m/s
Umin
Umean
Vmax
-0.87 m/s -0.01 m/s 0.50 m/s
-1.06 m/s -0.02 m/s 0.60 m/s
Vmin
Vmean
-0.62 m/s 0.00 m/s
-0.45 m/s 0.00 m/s
table 37
Velocities Bay Entry
Umax
Field Data
1.05 m/s
Mike21 BW 0.98 m/s
Umin
Umean
Vmax
-0.88 m/s -0.02 m/s 0.33 m/s
-0.93 m/s -0.01 m/s 0.47 m/s
Vmin
Vmean
-0.33 m/s 0.01 m/s
-0.42 m/s 0.00 m/s
table 38
Velocities Bay Middle
Umax
Umin
Umean
Vmax
Vmin
Vmean
Field Data
1.35 m/s -1.26 m/s -0.03 m/s 0.46 m/s -0.58 m/s -0.06 m/s
Mike21 BW 1.52 m/s -1.54 m/s 0.01 m/s 1.1 m/s -0.85 m/s -0.01 m/s
table 39
From table 37 it is observed how the change in velocity amplitudes are very small as would be
expected for the small change in wave direction. The magnitude of U has decreased slightly and the
magnitude of V has increased correspondingly simply due to the small change in the projection of
the incoming wave velocity vector to the domain coordinate system.
At station 2 in the bay middle the velocity magnitude of both U and V has decreased, which is due
to the simple fact that the wave action in this area (recall figure 131) is less than in Simulation2 and
flowingly also the wave induced velocities.
96
Again the same trend is observed in Station 3 assumable due to a slight decrease in wave height at
this location.
figure 133
figure 134
figure 135
As expected the direction scatter plots does not show any notable change in the directional velocity
distribution due to the small change in incident wave direction. In comparison it is remarkable how
an only 3-degree change in incident wave direction reduces deviation of model wave height
predictions at Station 2 with 9-10% without causing any notable change in prediction accuracy for
the two other Stations.
figure 136
figure 137
figure 138
As given by the average wave height properties the change in incident wave direction result in
minor changes in wave height in Station 1 and 3 and a larger change in Station 2. When examining
the time-series for the respective stations it is observed that while the adjusted incident wave
direction improves model results in terms of wave magnitude, it does not change the nature of the
waves occurring in the model domain. That is, the way incident and reflected waves are interaction
(producing the net signal) at the 3 stations remains rather unchanged.
As illustrated in the comparison between figure 130 and figure 131 the general wave conditions
does not change significantly by just changing the incident wave direction a few degrees. But the
experiences made in Simulation 6 confirms that 2 instruments inside the bay is not much to
97
sufficiently document the wave transformation processes occurring in a bay where both nonsymmetrical refraction and 2 dimensional wave reflection consists key governing processes.
98
4.3.7 Wednesday time-series (excl. porosity layer)
Simulation info: simulation 7
Data files: Wednesday v5
Wave-input: Time-series 1 (Wednesday)
Incident wave direction: 325
Bottom friction: C=32
Bathymetry: Domain 1
Porosity map: none
Upon assessing the models ability do simulate the energetic wave conditions occurring in the bay
Thursday morning it was decided to assess the model predictions for the much less wave conditions
occurring Wednesday afternoon. The model setup for this simulation is identical to Simulation 1
which the exception of the change in wave input condition at the offshore boundary. Also the still
water level has been raised 30 cm to account for the change in tidal elevation at this point in time.
Figure 139
99
Hm0
Hm0
Offshore
Bay Entry
Bay Middle
Field Data
0.77 m
0.64 m
0.60 m
Mike21 BW
0.80 m
0.61 m
1.02 m
Diff. [%]
4%
5%
70%
table 40
Mean Wave height
Field Data
Hm
0.48 m
Offshore
Bay Entry 0.36 m
Bay Middle 0.35 m
Mike21 BW
0.50 m
0.39 m
0.65 m
Diff. [%]
4%
8%
84%
Sig. Wave height
Field Data
Hsig
0.73 m
Offshore
Bay Entry 0.59 m
Bay Middle 0.55 m
table 41
Mike21 BW
0.73 m
0.59 m
1.01 m
Diff. [%]
0%
0%
84%
table 42
From the spectral plots and the wave height tables it is observed how there exists a virtually perfect
agreement between field data and model predictions for this location. It is recalled how previous
model simulations for the Thursday event (time-series 2) were often overestimating the wave height
offshore. The explanation for the improved prediction capability for this simulation could be that
the less energetic wave conditions in this simulation do not give rise to any significant contribution
from wave reflection to the signal at Station 1. But the fact that the input for this simulation is more
linear, and so does not cause the wave height to oscillate to the same degree due to its interaction
with its forced free higher order harmonics (recall figure 140), could also have contributed to the
improved model accuracy at this location.
At the bay entry it is noticed that the model
underestimates the spectral peak, while it slightly overestimates the higher order harmonics. It is
noticed that the predicted spectral peak is only about half the magnitude of the field data value.
However in spite of the notable disagreement in spectral amplitudes, there exists a very good
agreement in terms of H m0, Hm and Hsig.
In the bay middle the spectral peak this time is
overestimated by almost a factor 2. At the same time the higher order harmonics are significantly
overestimated just as with the equivalent model setup in Simulation 1.
100
figure 142
figure 141
From figure 141 it is observed how also for this Wednesday time-series there is a tremendous
amount of cross-sectional wave propagation taking place in the model just as experienced in
Simulation 1. Also from figure 142 it is observed how strong standing wave patterns are observed
in front of basically every single land boundary. It was expected that even more reflection would
occur inside the bay (both in the model and in the field) for time series 1 due to the longer wave
peak period and the smaller wave amplitude. However as wave reflection becomes of even greater
importance to the wave dynamical processes in the bay so does the models requirement to handle
the reflection correctly.
Velocities offshore
Umax
Field Data
0.53 m/s
Mike21 BW 0.38 m/s
Umin
Umean
-0.47 m/s 0.01 m/s
-0.42 m/s 0 m/s
Vmax
0.31 m/s
0.28 m/s
Vmin
Vmean
-0.29 m/s -0.01 m/s
-0.24 m/s 0.01 m/s
Vmax
0.15 m/s
0.31 m/s
Vmin
Vmean
-0.17 m/s -0.01 m/s
-0.27 m/s 0.00 m/s
Table 43
Velocities Bay Entry
Umax
Field Data
0.59 m/s
Mike21 BW 0.47 m/s
Umin
Umean
-0.48 m/s 0.01 m/s
-0.4 m/s 0.00 m/s
Table 44
Velocities Bay Middle
Umax
Umin
Umean
Field Data
1.01 m/s -0.85 m/s 0.00 m/s
Mike21 BW 0.61 m/s -0.55 m/s 0.00 m/s
Vmax
Vmin
Vmean
0.26 m/s -0.25 m/s -0.01 m/s
1.05 m/s -0.77 m/s -0.01 m/s
Table 45
101
Again it is noticed that the mean velocities are so small and insignificant that comparing them
would not be sensible.
It is observed that the amplitudes of the offshore velocities are in general good agreement with field
data. So are the U-amplitudes in the bay entry. The U-amplitudes in the bay middle are slightly
underestimated by the model. However the cross-sectional velocities in the bay entry and the bay
middle are significantly overestimated just as experienced in Simulation 1.
figure 143
figure 144
figure 145
From the velocity scatter plots it is seem how while there exists a good directional agreement at
offshore at Station 1, it is observed how the model overestimates the cross-sectional velocities due
to cross-sectional wave reflection. In the bay middle the significance of this phenomenon is all but
extreme. Again it can be concluded that the model overestimation of wave reflection from the cliffs
in the southern end of the bay also is of significant importance to wave conditions Wednesday
afternoon.
102
figure 146
figure 147
When comparing the time-series at Station 1 it is observed that the previous assumption that no
notable wave reflection is contributing to the wave signal at this location. However though this
seems to distort the signal the wave height characteristics are still sufficiently maintained.
figure 148
figure 149
When comparing the times-series in the bay middle it is observed how the simulated surface
elevation in the beginning only is approximately half of what is given by the measured field data.
However further into the time series the surface elevations are more in the same order of magnitude,
though it is observed how the simulated time-series is strongly affected by the extensive reflection
occurring in the model domain. This transition suggests that significant reflection occurs in the
domain and that Station 2 in this system is located near an anti-node causing amplification in wave
height. This could explain why the model underestimates the surface elevation for the first few
waves, as the reflected waves have not yet reached the station.
103
Figure 150
Figure 151
From the time-series comparison in the bay middle it is observed how the model predictions in the
beginning are in good agreement with field data but soon after starts to overestimate the wave
height at some locations due to an exaggerated predicted wave reflection especially from the
southern end of the Bay.
104
4.3.8 Wednesday time-series (incl. porosity layer)
Simulation info: simulation 8
Data files: Wednesday v7
Wave-input: Time-series 1 (Wednesday)
Incident wave direction: 325
Bottom friction: C=25
Bathymetry: Domain 1
Porosity map: 1
From the experiences with cross-sectional wave reflection when modeling Thursday’s wave-event,
it was not a surprise that the same phenomenon would occur for when modeling the wave event
from Wednesday. In order to account for this, the same porosity map as introduced in Simulation 2
is applied to this simulation in an attempt to improve model results.
figure 152
Hm0
Hm0
Offshore
Bay Entry
Bay Middle
Field Data
0.77 m
0.64 m
0.60 m
Mike21 BW
0.78 m
0.57 m
0.74 m
Diff. [%]
1%
11%
23%
table 46
105
Mean Wave height
Field Data
Hm
0.48 m
Offshore
Bay Entry 0.36 m
Bay Middle 0.35 m
Mike21 BW
0.48 m
0.35 m
0.44 m
Diff. [%]
0%
3%
35%
Sig. Wave height
Field Data
Hsig
0.73 m
Offshore
Bay Entry 0.59 m
Bay Middle 0.55 m
table 47
Mike21 BW
0.71
0.54
0.69
Diff. [%]
3%
8%
24%
table 48
From the spectral plots and the wave tables it is observed that a significant improvement to the
model performance is obtained by implementing a porosity map along the cliffs and rocky outcrops.
However from the spectral plots it is still observed that the spectral peak at Station 2 is
underestimated while it is overestimated at Station 3.
figure 153
figure 154
Intuitively one would expect the wave heights at Station 3 to be larger compared to Station 2 due to
shoaling. But previous simulations from Thursdays event has confirmed that the wave mechanical
processes in the bay is not that simple and there most likely are quasi-standing wave processes
occurring in the bay due to extensive wave reflection. Not only cross-sectional reflection from the
cliffs and rocky outcrops but also from the steep upper beach slope at the bottom of the bay. In
figure 154 it is observed how standing wave patterns from the especially the southern end of the bay
has been damped significantly compared to Simulation 7. However the damped reflection from the
south end has made another less energetic standing wave pattern become more visible. It is noticed
how waves seem to be reflected from the beach profile itself creating a standing wave pattern with
nodes and antinodes. From the Hm0 2D plot it looks like the model predicts an anti-node at the
106
location of Station 3 and a weak resemblance of some something that could look like a nodal point
at the location of Station 2. If this is correct it could explain why the spectral peak is amplified at
Station 3 and damped at Station 2. That is of course assuming that the same standing wave pattern
is not represented identically in the field data.
figure 155
figure 156
In figure 155 an envelope plot has been constructed from the surface elevation time-series along the
centerline of Waimea bay reaching from the wave generation line to the shoreline. Using Fourier
analysis a narrow band (f = 0.006 Hz) of frequencies centered on the peak frequency
corresponding to Tp=18.2 sec has be filtered out and presented in figure 156. Here it is indeed clear
that a standing wave is occurring from the beach shoreline. From video recordings simultaneously
recorded during Wednesdays time-series (time-series 1) it was easily visible how strong reflection
and near shore wave oscillation was occurring in the bay as also predicted by the model. However
given the fact that the wave conditions are highly reflective can make it very difficult to assess a
models capability to simulate a real-case system only having 2 reference points (field stations).
Even just a small phase-shift between the reflected waves in the field and those predicted by the
model can have a significant importance to the location of nodes and anti-nodes. Also the correct
prediction of the magnitude of the reflected wave brings in an additional aspect in the model
assessment.
Velocities offshore
Umax
Field Data
0.53 m/s
Mike21 BW 0.38 m/s
Umin
Umean
-0.47 m/s 0.01 m/s
-0.38 m/s 0 m/s
Vmax
0.31 m/s
0.27 m/s
Vmin
Vmean
-0.29 m/s -0.01 m/s
-0.16 m/s 0.01 m/s
Table 49
107
Velocities Bay Entry
Umax
Field Data
0.59 m/s
Mike21 BW 0.45 m/s
Umin
Umean
-0.48 m/s 0.01 m/s
-0.44 m/s 0.00 m/s
Vmax
0.15 m/s
0.16 m/s
Vmin
Vmean
-0.17 m/s -0.01 m/s
-0.14 m/s 0.00 m/s
Table 50
Velocities Bay Middle
Umax
Umin
Umean
Field Data
1.01 m/s -0.85 m/s 0.00 m/s
Mike21 BW 0.59 m/s -0.49 m/s 0.00 m/s
Vmax
Vmin
Vmean
0.26 m/s -0.25 m/s -0.01 m/s
0.32 m/s -0.36 m/s 0.00 m/s
Table 51
figure 157
figure 159
figure 158
In the velocity tables it is observed how the introduction of the porosity map helped correct the
cross-sectional velocity amplitudes at station 2 and 3. The same can be observed from the velocity
scatter plots, where a tremendous improvement has been obtained -Especially at Station 3 in the bay
middle.
figure 160
figure 161
108
figure 162
figure 164
figure 163
figure 165
From the time-series comparison for the 3 field stations it is observed how the implementation of
the porosity layer succeeds in damping the previously overestimated cross sectional reflection
affecting mostly the signal in Station 3. However even though this damping significantly improves
model results there still seems to be effects in the signal at Station 2 and 3 that does not seem to be
resolved well in the model.
109
5 Results Summary
Bay Entry
Bay Middle
Deviation [%]
Of fshore
100
80
60
40
20
Off shore
Bay Entry
Bay Middle
0
at
io
n
Si
1
m
ul
at
io
n
Si
2
m
ul
at
io
Si
n
3
m
ul
at
io
Si
n
4
m
ul
at
io
Si
n
5
m
ul
at
io
n
6
Si
Significant Wave height Comparison
(Wednesday, Time-series 1)
Significant Wave Height Comparison
(Thursday, Time-series 2)
m
ul
Deviation [%]
40
35
30
25
20
15
10
5
0
figure 166
8
7
ion ation
l
lat
u
u
Si m
Si m
figure 167
In the previous chapter it was observed that the Hm0, Hsig and Hm all shared the same trend for each
simulation, so in this result summary only a comparison of Hsig for the 8 simulations are listed in the
figures above. As mentioned previously, the initial model setup (using time-series 2 as input)
significantly overestimated the wave reflection from the steep cliffs in the domain. By applying a
porosity map at these locations, it was found that a great improvement of model results offshore and
especially at the bay middle was obtained. When applying a new wave input time-series (timeseries 1) to the model domain the same problem with overestimated wave reflection was
experienced and a similar improvement was made with respect to the bay middle when applying the
model domain to the different wave conditions as given by time-series 1.
110
figure 168
figure 169
It was found that Station 2 was located in the proximity of a small zone with smaller wave height
created by the sheltering effect of the adjacent reef located off the northern end of the bay. In
Simulation 7 and 8 (using time-series 1) Station 2 is located in the middle of this sheltered zone and
the wave height prediction is in good agreement with field data. But in Simulation 1 to 6 (using
time-series 2) Station 2 is predicted to be located on the rim of the sheltered zone and the wave
height prediction is notably overestimated. This suggests that the predicted size and/or location
might be slightly inaccurate. A number of simulation attempts were carried out in order to improve
this prediction, but it was only by altering the wave direction (+3 N) that Station 2 became more
included into the now expanded sheltered zone (due to the increased sheltering effect by the reef)
and the wave height prediction was improved significantly.
111
figure 170 – Simulation 7
figure 171 – Simulation 2
Another phenomenon noticed in all simulations was the appearance of a standing wave pattern
occurring along the centerline of the bay. The bay entry is located approximately 560 m from the
wave generation line and from figure 170 and figure 171 it is observed that the standing wave
pattern extend beyond this point an so effecting both the predictions at Station 2 and Station 3.
While the wave height prediction at Station 3 seems in good agreement when using time-series 2
and applying a porosity map (Simulation 2-6), the wave height is notably overestimated when using
time-series 1 even when applying a porosity map (Simulation 8). The best suggestion would be that
the model in this case either overestimates the wave reflection occurring from the beach slope,
which will be further discussed in the next chapter.
112
figure 172
In figure 172 a brief wave height comparison has been carried out between the Mike21 Boussinesq
wave model (Mike21BW) and the Mike21 Spectral Wave model (Mike21SW). The Boundary
condition used for the spectral wave model has been obtained from the wave buoy located 5.6 km
offshore Waimea bay. While the application of a spectral model is not part of this study, it was
found interesting to draw the comparison, as the spectral model is not capable of handling wave
reflection (as opposed to BW). It is very interesting to see while both models capture the sheltering
effect from the reef (at the north end), the spectral wave model does not capture the local effects
due to significant wave reflection occurring from the landward boundary, which is of mayor
importance to the predicted wave height distribution inside the bay.
Umean Comparison, Time series 2
Offshore Bay Entry Bay Middle
Field Data -0,01 m/s -0,02 m/s -0,03 m/s
Simulation 1 -0,05 m/s -0,01 m/s 0,01 m/s
Simulation 2 -0,03 m/s -0,01 m/s -0,01 m/s
Simulation 3 -0,03 m/s 0.00 m/s 0,02 m/s
Simulation 4 -0,03 m/s -0,01 m/s 0,01 m/s
Simulation 5 0.00 m/s 0.00 m/s 0,01 m/s
Simulation 6 -0,02 m/s -0,01 m/s 0,01 m/s
table 52
Vmean Comparison, Time series 2
Offshore Bay Entry
Field Data 0.00 m/s 0,01 m/s
Simulation 1 0,01 m/s 0,01 m/s
Simulation 2 0,01 m/s 0.00 m/s
Simulation 3 0,01 m/s 0.00 m/s
Simulation 4 0,01 m/s 0.00 m/s
Simulation 5 0.00 m/s -0,01 m/s
Simulation 6 0.00 m/s 0.00 m/s
Bay Middle
-0,06 m/s
-0,01 m/s
-0,01 m/s
0.00 m/s
0.00 m/s
-0,02 m/s
-0,01 m/s
table 53
113
Umean Comparison, Time series 1
Offshore Bay Entry
Field Data 0,01 m/s 0,01 m/s
Simulation 7 0,00 m/s 0,00 m/s
Simulation 8 0,00 m/s 0,00 m/s
Bay Middle
0,00 m/s
0,00 m/s
0,00 m/s
table 54
Vmean Comparison, Time series 1
Offshore Bay Entry Bay Middle
Field Data -0,01 m/s -0,01 m/s -0,01 m/s
Simulation 7 0,01 m/s 0,00 m/s -0,01 m/s
Simulation 8 0,01 m/s 0,00 m/s 0,00 m/s
table 55
As discussed in the previous chapter and listed in the tables above, the wave driven currents at
Station 1, 2 and 3 are very small and insignificant when modeling time-series 2 and basically nonexistent when modeling time-series 1. The explanation for the very small wave driven currents in
the bay was simply due to the lack of widespread wave breaking, which almost only occurred at the
reef at the northern end and very close to the shoreline. Wave breaking only occurred with the
arrival of a wave-set followed by several minutes of little or no wave breaking. As a result the wave
driven currents in most part of the bay (even the most energetic part of the wave event) must be
considered to be fairly small. At the 3 stations it was found that the model predictions at these
locations were mostly in the correct order of magnitude but due to the extremely small values very
sensitive to even small changes in the model setup.
114
figure 173 - Wave-Driven Current Plot (Simulation 2)
From figure 173 it is observed how the most dominant current patterns occurs along northern tip of
the bay due to pronounced wave breaking in this area. Also a less significant pattern appears along
the south side of the bay and also a small rip current seems to be predicted at the southern end of the
Waimea bay beach. However as both Station 2 and 3 are located outside any of these areas of
interest it is not possible in any way to assess the performance of the model predictions of the wave
driven currents.
115
6 Conclusion
At first it seemed like an easy task to apply a Boussinesq model to simulate the waves and wave
driven currents in Waimea Bay. From the results obtained in the preliminary study it was evident
that the MIKE21 BW model held an excellent capability of simulating both wave breaking,
shoaling, refraction, diffraction and wave induced current circulation. So from a theoretical point of
view, the usually larger deviations experienced during the Waimea Bay model simulations might
initially seem less impressive.
But once realizing the complexity and sensitivity of the wave dynamical processes occurring within
the domain combined with the potential sources of error associated with this study, it is in fact
excessively impressive that the Boussinesq model actually succeed in providing a fair to good
prediction of the wave and wave induced current conditions for all 3 stations.
The wave conditions inside the bay during simulation were far from monotonous and largely
influenced by the combined effects from non-uniform wave refraction and excessive wave
reflection from the steep cliffs and rocky outcrops and also from the beach itself.
When modeling time-series 2 it was found that wave height predictions could be greatly improved
by just a small change in incident wave direction. In comparison it should be taken into
consideration that the ADCP compass (from which the wave direction is obtained) is only accurate
within 3-4. Secondly, even though assuming the incident wave train to be unidirectional is a fair
assumption for the given swell event, the velocity scatter plots does show some directional
spreading (recall figure 44 and figure 45) within the magnitude of 2-5 degrees. Additionally it
should be recalled that the position accuracy of the handheld GPS used to place the instruments
inside the bay is only within 5-10m (sometimes even worse). For reference it should be considered
that if the true position of Station 2 in fact was located 47 m North East of its assumed location, this
alone would result in a perfect agreement in wave height prediction for Simulation 2 (recall figure
168). With this information in mind it would seem harsh to criticize the model for overestimating
the wave height by 25% in at this location.
Another problem during simulations was the models tendency to exaggerate the magnitude of the
wave reflection occurring from the steep cliffs and rocky outcrops in the domain, which is most
likely due to the limitations in the method in which the model handles breaking waves. As
mentioned previously, the breaker module is designed to handling spilling breakers, which assumes
a gradual dissipation of wave energy. The preliminary study has shown that the breaker model also
116
can be successfully used in modeling plunging breakers, which occurs on the reef along the north
entrance of the bay. But the breaker model is not designed to handle the rapid energy dissipation
occurring at the powerful collapsing breakers occurring along the steep cliffs and rocky outcrops in
the domain.
(Thursday AM, April 20. 2006)
(Thursday AM, April 20. 2006)
The very narrow surf zone in front of the cliffs and rocky The Waimea Bay shore-break.
outcrops in the south entrance of the bay.
In addition the cliffs and outcrops contain rugged features that contribute to even more extensive
dissipation but are not resolved in the model bathymetry due to the current 2x2 m grid resolution.
It was found that introducing a porosity map could damp the exaggerated wave reflection from the
cliffs and rocky outcrops, but as concluded from the time series study the porosity map only
monotonously damps the wave reflection occurring from these locations. Hence it does not capture
the local wave reflection effects occurring at these areas, where full reflection might occur at some
locations and perhaps not at all at other locations due to the wave response to local variations and
small-scale features in the bathymetry.
Also along the shoreline of Waimea bay significant wave reflection has also been shown to occur in
the model for both time-series 1 and 2. From simultaneously recorded video footage (Appendix
VIII) it is easily observed that pronounced wave reflection at this location also occur in the field but
from only 2 field stations inside the bay it is not possible to fully clarify if a standing wave pattern
is apparent in the bay and if this in any way compares well with model predictions. It can only be
concluded that it was possible to obtain a wave height prediction deviation at Station 3 of less than
5% when simulating time-series 2, while the prediction accuracy using the same model setup was
more than 20% when simulating time-series 1. This could be explained by comparing the dominant
breaker type occurring near the shoreline for the two simulations using the previously mentioned
117
surf similarity parameter. First it should be advised that while the average beach slope for the entire
domain is 0.04 it is 0.06 close to the shoreline. If drawing the rough assumption that the mean wave
height measured at Station 3 equals the wave breaker height Hb, and using the peak period Tp to
calculate the deepwater wavelength L0, the surf similarity parameter (close to the shoreline) will be
1.2 for time-series 2 suggesting a plunging breaker and 2.3 for time-series 1 suggesting a surging or
collapsing breaker. Recorded video footage during the two time-series also at least qualitatively
confirms this assumption. But it could explain why the wave reflection is overestimated in timeseries 1 as the breaker module does not sufficiently well simulate the energy dissipation
characteristics of the collapsing wave, while the wave energy dissipation characteristics of the
plunging wave breakers occurring in time-series 2 are probably more adequately described by the
current breaker module.
Just from a quick glance at any of the H m0-plots produced in this study, it is obvious that 2
instruments inside the bay is nowhere near sufficient to properly describe the wave conditions
inside Waimea Bay and provide a adequate base for comparison and assessment of the model
predictions for this domain.
In spite of this, The Mike21 BW model was actually able to produce results that agreed fairly well
with field measurements even in a direct time-series comparison. It is the author’s opinion that the
most dominant source of model inaccuracy was due to the way partial wave-reflection was handled
in the model. In some cases due to the insufficient wave dissipation when modeling collapsing
breakers and also due to the necessity to introduce artificial damping in areas with highly steep and
uneven bathymetry such as in front of the steep cliffs and rocky outcrops. Unfortunately it is hard
(from only two field stations inside the bay) to quantify how much the models prediction of
reflection actually deviates from what is actually occurring in the bay. As mentioned previously the
deviation could easily be limited to small phase shifts between the incident and reflected wave train
and local effects in some problematic areas of the domain.
Again the potential complications and sources of inaccuracies associated with extracting the field
data and problems in correctly resolving the near shore bathymetry (important to wave reflection)
could almost be a study by its own.
But even though hindsight might suggest that some parts of this Waimea Bay study could have been
carried out differently (especially with respect to the insufficient field instrumentation), the
118
agreement between field data and model results has been satisfying not to mention that the
educational gains has been more than significant. This cover a wide range of topics such as how to
carry out a full-scale field campaign in the coastal zone, post-processing and interpreting the raw
field data to how to apply a state-of-the-art Boussinesq wave model to simulate the highly complex
wave environment in Waimea Bay. So once fully realizing the complexity and challenging aspects
of this case study, it is not hard to conclude that, based on the obtained project results as well as the
associated general and case-specific educational gains, the project has been a tremendous success.
Simon Brandi Mortensen
119
7 References
DHI Software, 2004, “Mike21 Boussinesq Wave Modules User Guide”, DHI Water and
Environment, Hørsholm, Denmark
Kubota, S, Mizuguchi, M and Takezawa, M, 1989,”Reflection from swash zone on natural
beaches”, Coastal Eng., 23, 570-583.
Madsen, P A, 2003, “An introduction to higher order spectral analysis and stochastic evolution
equations”, Lecture notes, Technical University of Denmark.
Madsen, P A and Sørensen, O R, 1992, “A new form for the Boussinesq equations with improved
linear dispersion characteristics. Part 2: A slowl- varying Bathymetry ”. Coastal Eng., 18, 183204.
Madsen, P A, Sørensen, O R and Scäffer, H A, 1997a, “Surf zone dynamics simulated by a
Boussinesq type model. Part I: Model description and cross-shore motion of regular waves”.
Coastal Eng., 32, 255-288.
Madsen, P A, Sørensen, O R and Scäffer, H A, 1997b, “Surf zone dynamics simulated by a
Boussinesq type model. Part II: Surf beat and swash zone oscillations for wave groups and
irregular waves”. Coastal Eng., 32, 289-320.
Nielsen, P, 2003, Teaching notes on coastal and estuarine processes, Department of Civil
Engineering, University of Queensland
Smith, S W, 1999, The Scientist and Engineer’s Guide to Digital Signal Processing, 2nd Edition,
California Technical Publishing, San Diego California.
Sørensen, O R, Scäffer, H A and Madsen, P A, 1998, “Surf zone dynamics simulated by a
Boussinesq type model. Part III: Wave-induced horizontal near shore circulations”. Coastal Eng.,
33, 155-176.
Web sites:
Joint Airborne Lidar Bathymetry Technical Center of Expertise (JALBTCE):
http://shoals.sam.usace.army.mil
Nortek (manufacturer of Current Profilers)
http://www.nortek-as.com
Teledyne RD Instruments – Manufacturer of ADCP
http://www.rdinstruments.com
The Coastal Data Information Program (Historic Waimea Buoy Data)
http://cdip.ucsd.edu/?pub=nonpub&stn=206&stream=p1&nav=historic&sub=data
120
8 Appendices
Appendix I
Matlab code used to extract velocity time-series for a chosen time frame from the ADCP data.
Appendix II
Matlab code to extract a selected surface elevation time-series from the ADCP
Appendix III
Matlab code use to calculate wave height, spectral peak period and wave direction for the 48-hour
field dataset.
Appendix IV
Matlab code used to extrat selected velocity and surface elevation time-series from the AquaDopps.
Appendix V
Mablab code used for conducting time-series analysis of model results and calculate Hsig, Hm, Um,
Vm, Umax /Umin and Vmax/Vmin.
Appendix VI
Matlab code used for spectral analysis of model results.
Appendix VII
Matlab code used to calculate wave height distribution and a filtered envelope plot along the
centerline of Waimea bay.
Appendix VIII
3 CD-RWs containing:





3D Animation of Simulation 2.
Time-stamped video clip recorded during time-series 1
Time-stamped video clip recorded during time-series 2 #1
Time-stamped video clip recorded during time-series 2 #2
Time-stamped video clip of the swash zone recorded 30 minutes after time-series 2
121
Appendix I
clear all
close all
u_input=load('u_velocity.txt');
v_input=load('v_velocity.txt');
w_input=load('w_velocity.txt');
time_vector=load('ADCP_timevector.txt')';
u_data =u_input(:,2)';
v_data =v_input(:,2)';
w_data =w_input(:,2)';
Start_date=datestr([2006 04 19 08 30 59],0);
time_origo=datenum(2006,04,19,08,59,09);
time=time_vector;
%Velocity direction during selected timeseries
%............................................%
% Selecting the surface elevation timeseries %
%............................................%
% Select slice
start=find(time_vector<=datenum(2006,04,20,09,25,00),1,'last');
finish=find(time_vector<=datenum(2006,04,20,09,55,00),1,'last');
duration=30*60;
% U-DATA
%Inforcing a timestep dt of 1 sec for the selected slice using
%interpolation.
time_num=time_vector*24*3600;
new_time(1)=time_num(start);
u_slice(1)=u_data(start);
for ii=2:duration;
new_time(ii)=new_time(ii-1)+1;
Find_pre_time=find(time_num<new_time(ii),1,'last');
Find_next_time=find(time_num>=new_time(ii),1,'first');
Pre_time=time_num(Find_pre_time);
Next_time=time_num(Find_next_time);
Pre_u=u_data(Find_pre_time);
Next_u=u_data(Find_next_time);
u_slice(ii)=(Next_u-Pre_u)/(Next_time-Pre_time)*(1-(Pre_time-new_time(ii-1)))+Pre_u;
end
new_time(duration+1)=new_time(duration)+1;
u_slice(duration+1)=u_data(finish);
% Control correct interpolation scheme
figure
122
plot(time_vector(start:finish),u_data(start:finish),'-o')
hold on
plot(new_time/(24*3600),u_slice,'x-')
figure
plot(time,u_data)
title('East-bound depth averaged velocity U0');
xlabel('time from 8.31 AM, Wednesday morning [hours]');
ylabel('velocity [m/s]');
saveas(gcf,'u_velocity', 'jpeg')
figure
plot(time_vector(start:finish),u_data(start:finish))
title([char('East-bound depth averaged velocity U0','Starting time: ',num2str(datestr(time_vector(start),0)),'
End time: ',num2str(datestr(time_vector(finish),0)),'')]);
xlabel('time [minutes]');
ylabel('velocity [m/s]');
saveas(gcf,'u_velocity_slice', 'jpeg')
u_velocity_slice=u_slice';
save u_slice_wednesday1615.txt u_velocity_slice -ASCII
save u_slice_wednesday1615.m u_velocity_slice -mat
% V-DATA
%Inforcing a timestep dt of 1 sec for the selected slice using
%interpolation.
v_slice(1)=v_data(start);
for ii=2:duration;
Find_pre_time=find(time_num<new_time(ii),1,'last');
Find_next_time=find(time_num>=new_time(ii),1,'first');
Pre_time=time_num(Find_pre_time);
Next_time=time_num(Find_next_time);
Pre_v=v_data(Find_pre_time);
Next_v=v_data(Find_next_time);
v_slice(ii)=(Next_v-Pre_v)/(Next_time-Pre_time)*(1-(Pre_time-new_time(ii-1)))+Pre_v;
end
v_slice(duration+1)=v_data(finish);
% Control correct interpolation scheme
figure
plot(time_vector(start:finish),v_data(start:finish),'-o')
hold on
plot(new_time/(24*3600),v_slice,'x-')
figure
plot(time,v_data)
title('North-bound depth averaged velocity V0');
xlabel('time from 8.31 AM, Wednesday morning [hours]');
ylabel('velocity [m/s]');
saveas(gcf,'v_velocity', 'jpeg')
figure
123
plot(time_vector(start:finish),v_data(start:finish))
title([char('North-bound depth averaged velocity V0','Starting time: ',num2str(datestr(time_vector(start),0)),'
End time: ',num2str(datestr(time_vector(finish),0)),'')]);
xlabel('time [minutes]');
ylabel('velocity [m/s]');
saveas(gcf,'v_velocity_slice', 'jpeg')
v_velocity_slice=v_slice';
save v_slice_wednesday1615.txt v_velocity_slice -ASCII
save v_slice_wednesday1615.m v_velocity_slice -mat
figure
plot(time,w_data)
title('Vertical depth averaged velocity W0');
xlabel('time from 8.31 AM, Wednesday morning [hours]');
ylabel('velocity [m/s]');
saveas(gcf,'w_velocity_wednesday1615', 'jpeg')
%
% fitfunction=fit(u_data(start:finish)',v_data(start:finish)','poly1')
% Mag_correction=10.4; % correction for magnetic north
% wave_dir=abs(atan(fitfunction.p1)/pi*180)+270+Mag_correction;
%Determining Wave Direction
u=u_velocity_slice;
v=v_velocity_slice;
Mag_correction=-(10.4)*pi/180; % correction for magnetic north
u2=u*cos(Mag_correction)-v*sin(Mag_correction);
v2=v*cos(Mag_correction)+u*sin(Mag_correction);
fitfunction=fit(u2,v2,'poly1')
wave_dir=abs(atan2(1,fitfunction.p1)/pi*180)+180
figure
plot(u2,v2,'.')
hold on
plot(u2,fitfunction.p1*u2,'r')
legend('Velocity Scatter Plot','Mean wave direction')
title(['Thursday 20/04-06 9.25-9.55, \theta_m= ',num2str(round(wave_dir)),'\circ']);
xlabel('U0 [m/s], east-bound velocity');
ylabel('V0 [m/s], north-bound velocity');
axis([-2 2 -2 2])
save u_rotated_thursday925.txt u2 -ASCII
save v_rotated_thursday925.txt v2 -ASCII
124
Appendix II
Matlab code to extract a selected surface elevation time-series from the ADCP
clear all
close all
tic
g=9.81;
rho=1000;
ratio=10^4/(rho*g); %conversion from dbar to pressure in [m]
data=load('ADCP_pressure.txt');
time_vector=data(:,1);
pressure=ratio*data(:,2);
time_origo=time_vector(1);
%Input remarks
adcp_offset=0.30;
dt=1;
time=[0:1:length(pressure)-1]*dt;
%Filtering out the tidal oscilation
average_length=1000;
waterdepth_fraction(1)=mean(pressure(1:average_length));
waterdepth=ones(1,average_length)*waterdepth_fraction(1);
for i=1:fix((length(pressure))/average_length)-1
waterdepth_fraction(i+1)=mean(pressure(i*average_length:(i+1)*average_length));
waterdepth_vector=ones(1,average_length)*waterdepth_fraction(i);
waterdepth=[waterdepth waterdepth_vector];
end
%What is left after the last batch is assigned same waterlevel
av_length=fix((length(pressure))/average_length)*average_length;
rest=length(pressure)-av_length; %last part not included in the averaging
rest_vector=ones(1,rest)*waterdepth_fraction(fix((length(pressure))/average_length));
waterdepth=[waterdepth rest_vector];
delta_p=pressure'-waterdepth;
bins=2^13
% Compute spectrum
fn=1/(dt*2);
Y1=fft(delta_p,bins);
p_spectrum=((real(Y1(1:end)).^2+imag(Y1(1:end)).^2).^0.5)/bins*2;
p_spectrum(1)=p_spectrum(1)/2;
f=[0:bins/2]/bins*2*fn;
Find_max_total=find(p_spectrum==max(p_spectrum),1);
Tpeak_total=(f(Find_max_total))^(-1);
%plot spectrum
figure
plot(f,p_spectrum(1:length(f)))
125
title([char('Wave Powerspectrum (Entire timeseries)',' Tpeak: ',num2str(Tpeak_total),'Starting time:
',num2str(datestr(time_origo,0)),' End time:
',num2str(datestr(time_origo+length(pressure)/(24*3600),0)),'')],'HorizontalAlignment','center');
xlabel('frequency [Hz]');
ylabel('');
saveas(gcf,'Total_spectrum', 'jpeg')
%plotting figures
figure
plot(time,pressure,'r')
hold on
plot(time, waterdepth,'b')
title(['Starting time: ',num2str(datestr(time_origo,0)),' End time:
',num2str(datestr(time_origo+length(pressure)/(24*3600),0)),'']);
xlabel('time [seconds]');
ylabel('meters');
legend('ADCP pressure', 'Tidal motion');
saveas(gcf,'ADCP_pressure', 'jpeg')
%............................................%
% Selecting the surface elevation timeseries %
%............................................%
% Select slice
start=find(time_vector<=datenum(2006,04,20,09,25,00),1,'last');
finish=find(time_vector<=datenum(2006,04,20,09,55,00),1,'last');
duration=30*60;
%Inforcing a timestep dt of 1 sec for the selected slice using
%interpolation.
time_num=time_vector*24*3600;
new_time(1)=time_num(start);
p_slice(1)=delta_p(start);
for ii=2:duration;
new_time(ii)=new_time(ii-1)+1;
Find_pre_time=find(time_num<new_time(ii),1,'last');
Find_next_time=find(time_num>=new_time(ii),1,'first');
Pre_time=time_num(Find_pre_time);
Next_time=time_num(Find_next_time);
Pre_pressure=delta_p(Find_pre_time);
Next_pressure=delta_p(Find_next_time);
p_slice(ii)=(Next_pressure-Pre_pressure)/(Next_time-Pre_time)*(1-(Pre_time-new_time(ii1)))+Pre_pressure;
end
new_time(duration+1)=new_time(duration)+1;
p_slice(duration+1)=delta_p(finish);
figure
plot(time_vector(start:finish),delta_p(start:finish),'-o')
hold on
plot(new_time/(24*3600),p_slice,'x-')
126
%Slice up the selected surface elevation file
time_slice=new_time;
%p_slice=delta_p(begin:stop)';
relative_t=[0:dt:length(p_slice)-1];
% Compute spectrum of selected slice
bins_slice=2^11;
Y=fft(p_slice,bins_slice);
p_slice_spectrum=((real(Y(1:end)).^2+imag(Y(1:end)).^2).^0.5)/bins*2;
p_slice_spectrum(1)=p_slice_spectrum(1)/2;
f_slice=[0:bins_slice/2]/bins_slice*2*fn;
Find_max=find(p_slice_spectrum==max(p_slice_spectrum),1);
Tpeak=(f_slice(Find_max))^(-1);
water_level=mean(waterdepth(start:finish))+adcp_offset;
% Plot spectrum of selected slice
figure
plot(f_slice,p_slice_spectrum(1:length(f_slice)))
title([char('Wave Powerspectrum (Selected timeseries) Tpeak: ',num2str(Tpeak),'Starting time:
',num2str(datestr(new_time(1)/(24*3600),0)),' End time: ',num2str(datestr(new_time(end)/(24*3600),0)),' ')]);
xlabel('frequency [Hz]');
ylabel('');
saveas(gcf,'wavespectrum_slice', 'jpeg')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Syntesize Untransformed Wavespectrum: Frequency space => Time Series
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Re_Y1=real(Y(1))/bins_slice;
Re_Y2=real(Y(2:end-1))/(bins_slice/2); %Correcting the real parts.
Re_Y3=real(Y(bins_slice/2))/bins_slice;
Re_Y=[Re_Y1 Re_Y2 Re_Y3];
Im_Y=-imag(Y(1:end))/(bins_slice/2);
%Create low-pass filter
fmin=0.1928; %Hz
bandwidth=1/(fmin^(-1)-2); %Hz
Find_cutoff=find(f_slice>=fmin & f_slice<=bandwidth);
L_band=[0:length(Find_cutoff)-1]*pi/(2*length(Find_cutoff));
Lowpass_fx=(cos(L_band).^2)';
for kk=1:Find_cutoff(1)
merged(kk,:)=Re_Y(kk).*cos(2*pi*f_slice(kk)*relative_t)+Im_Y(kk)*sin(2*pi*f_slice(kk)*relative_t);
end
% %Apply Lowpass filter
%
% jj=1;
% for pp=Find_cutoff(1):Find_cutoff(end)
% merged(pp,:)=merged(pp,:)*Lowpass_fx(jj);
% jj=jj+1;
% end
127
reform_wave=sum(merged);
% CONTROLING CORRECT FOURIER ANALYSIS
figure
plot(time_slice,p_slice,'r')
title('Original signal compared to fourier synthesis');
hold on
plot(time_slice,reform_wave+2.5,'b')
saveas(gcf,'slice_control', 'jpeg')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Syntesize The transformed Wavespectrum: Frequency space => Time Series
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%calc k
omega=f_slice*2*pi;
L0=g*f_slice.^(-2)/(2*pi);
k_guess=2*pi*L0.^(-1);
for pp=1:length(f_slice)
delta=1;
while delta>10^(-6)
L(pp)=L0(pp)*tanh(k_guess(pp)*water_level);
k(pp)=2*pi/L(pp);
delta=abs(k(pp)^2-k_guess(pp)^2);
k_guess(pp)=k(pp);
end
end
GN_function=(cosh(k*water_level)./cosh(k*adcp_offset))';
figure
plot(k,GN_function)
title('The "pressure=>surface elevation transfer function');
xlabel('k');
ylabel('GN');
saveas(gcf,'GN_function', 'fig')
%plot(k)
merged_trans(1,:)=merged(1,:)
for vv=2:Find_cutoff(1)
merged_trans(vv,:)=merged(vv,:)*GN_function(vv);
end
surf_elevation=sum(merged_trans);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Wave Stats
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f_band=0.01;
df=(bins_slice*2*fn)^(-1);
bin_size=round(f_band/df);
f_slice=[0:bins_slice/2]/bins_slice*2*fn;
bin_number=fix(length(f_slice)/bin_size);
128
f_new=[0.5:bin_number-0.5]*f_band;
Y2=fft(surf_elevation,bins_slice);
p_slice_spectrum=((real(Y2(1:end)).^2+imag(Y2(1:end)).^2).^0.5)/bins_slice*2;
p_slice_spectrum(1)=p_slice_spectrum(1)/2;
for cc=1:bin_number
new_spect(cc)=mean(p_slice_spectrum((cc-1)*bin_size+1:cc*bin_size));
end
Tpeak=f_new(find(new_spect==max(new_spect)))^(-1);
%locate upcrossings
% surf_elevation=surf_elevation_log(47,:);
for ss =1:length(surf_elevation)-1
if surf_elevation(ss)<0 & surf_elevation(ss+1)>0;
nm_log(ss)=ss;
end
end
Finder=find(nm_log>0);
% find wavecrest and wavetroughs
tt=1;
for tt = 1:length(Finder)-1
crest(tt)=max(surf_elevation(Finder(tt):Finder(tt+1)));
trough(tt)=min(surf_elevation(Finder(tt):Finder(tt+1)));
end
Waveheight=crest-trough;
Hm=mean(Waveheight);
sort_fx=sort(Waveheight,'descend');
length_vector=round(length(sort_fx)/3);
Hsig=mean(sort_fx(1:length_vector));
figure
plot(f_new,new_spect)
title([char('Powerspectrum: [16.15-16.45] Wednesday April 19 2006')]);
xlabel('frequency [Hz]');
ylabel('m^2/Hz');
saveas(gcf,'wavespectrum_slice', 'jpeg')
axis([0 0.5 0 0.03])
save powerspectrum_thursday925.txt new_spect -ASCII
save F_new.txt f_new -ASCII
%Log values
Tpeak
Hm
Hsig
Hmax=max(Waveheight)
figure
plot(surf_elevation,'-.')
hold on
plot(p_slice,'r')
title(['Starting time: ',num2str(datestr(new_time(1)/(24*3600),0)),' End time:
',num2str(datestr(new_time(end)/(24*3600),0)),' ']);
xlabel('time [seconds]');
129
ylabel('meters');
axis([0 1800 -2 2.2])
legend('\eta(t), f = 0 - 0.19 Hz',' p^+(t), ADCP)');
saveas(gcf,'surfelevation_slice', 'fig')
surf_elevation=surf_elevation';
save input_surfelevation_thursday925_955.txt surf_elevation -ASCII
%save surf_slice_thursday925_955.m surf_elevation -mat
disp(['Starting time: ',num2str(datestr(new_time(1)/(24*3600),0)),''])
disp(['Ending time: ',num2str(datestr(new_time(end)/(24*3600),0)),''])
disp(['Number of timesteps: ',num2str(length(p_slice)),''])
disp(['Average Waterlevel: ',num2str(water_level),'meters'])
disp(['Waterlevel diff: ',num2str(waterdepth(start)-waterdepth(finish)),'meter'])
disp(['Wave peak period: ',num2str(Tpeak),'sec'])
p_slice=p_slice';
%Save pressure
save pressure_ADCP_thursday925.txt p_slice -ASCII
figure
plot(surf_elevation,'-.','LineWidth',1.5)
hold on
plot(p_slice,'r','LineWidth',1.5)
title('Transformation from intrument measured pressure to surface elevation (ADCP)');
xlabel('time [seconds]');
ylabel('meters');
legend('\eta(t), f = 0 - 0.19 Hz',' p^+(t)');
saveas(gcf,'surfelevation_slice', 'fig')
axis([100 200 -1.5 2])
130
Appendix III
Matlab code use to calculate wave height, spectral peak period and wave direction for the 48-hour
field dataset.
clear all
close all
g=9.81;
rho=1024;
ratio=10^4/(rho*g); %conversion from dbar to pressure in [m]
pressure=ratio*load('pressure.txt');
u_input=load('u_velocity.txt');
v_input=load('v_velocity.txt');
u_data =u_input(:,2)';
v_data =v_input(:,2)';
time_origo=datenum(2006,04,19,08,59,09);
%Input remarks
adcp_offset=0.30;
dt=1;
%resolution for spectrum calc;
bins_slice=2^12;
f_band=0.01;
time=[0:1:length(pressure)-1]*dt;
%Filtering out the tidal oscilation
average_length=1000;
waterdepth_fraction(1)=mean(pressure(1:average_length));
waterdepth=ones(1,average_length)*waterdepth_fraction(1);
for i=1:fix((length(pressure))/average_length)-1
waterdepth_fraction(i+1)=mean(pressure(i*average_length:(i+1)*average_length));
waterdepth_vector=ones(1,average_length)*waterdepth_fraction(i);
waterdepth=[waterdepth waterdepth_vector];
end
%What is left after the last batch is assigned same waterlevel
av_length=fix((length(pressure))/average_length)*average_length;
rest=length(pressure)-av_length; %last part not included in the averaging
rest_vector=ones(1,rest)*waterdepth_fraction(fix((length(pressure))/average_length));
waterdepth=[waterdepth rest_vector];
delta_p=pressure'-waterdepth;
%Number of 30 min segments
segment=60 %minutes
seg_num=fix(length(delta_p)/(segment*60))
tic
for bb = 1:seg_num
bb % show progress
131
%............................................%
% Selecting the surface elevation timeseries %
%............................................%
%Slice up the selected surface elevation file
duration=segment*60;
begin=1+(bb-1)*duration
stop=begin+duration
time_slice=time(begin:stop);
p_slice=delta_p(begin:stop)';
u_slice=u_data(begin:stop)';
v_slice=v_data(begin:stop)';
fitfunction=fit(u_slice,v_slice,'poly1')
Mag_correction=10.4; % correction for magnetic north
wave_dir(bb)=abs(atan(fitfunction.p1)/pi*180)+270+Mag_correction;
relative_t=[0:dt:length(p_slice)-1];
% Compute spectrum of selected slice
fn=1/(dt*2);
df=(bins_slice*2*fn)^(-1);
bin_size=round(f_band/df);
f_slice=[0:bins_slice/2]/bins_slice*2*fn;
bin_number=fix(length(f_slice)/bin_size);
f_new=[0.5:bin_number-0.5]*f_band;
Y=fft(p_slice,bins_slice);
p_slice_spectrum=((real(Y(1:end)).^2+imag(Y(1:end)).^2).^0.5)/bins_slice*2;
p_slice_spectrum(1)=p_slice_spectrum(1)/2;
for cc=1:bin_number
new_spect(cc)=mean(p_slice_spectrum((cc-1)*bin_size+1:cc*bin_size));
end
Tpeak=f_new(find(new_spect==max(new_spect)))^(-1);
p_slice_spectrum_log(bb,:)=p_slice_spectrum;
%f_slice=[0:bins_slice/2]/bins_slice*2*fn;
%
%
Find_max=find(p_slice_spectrum==max(p_slice_spectrum),1);
Tpeak=(f_slice(Find_max))^(-1);
water_level=mean(waterdepth(begin:stop))+adcp_offset;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Syntesize The transformed Wavespectrum: Frequency space => Time Series
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Re_Y1=real(Y(1))/bins_slice;
Re_Y2=real(Y(2:end-1))/(bins_slice/2); %Correcting the real parts.
Re_Y3=real(Y(bins_slice/2))/bins_slice;
Re_Y=[Re_Y1' Re_Y2' Re_Y3'];
132
Im_Y=-imag(Y(1:end))/(bins_slice/2);
kk=1;
for kk=1:bins_slice/2
merged(kk,:)=Re_Y(kk).*cos(2*pi*f_slice(kk)*relative_t)+Im_Y(kk)*sin(2*pi*f_slice(kk)*relative_t);
end
%Frequency cutt-off
fmin=0.1928; %Hz
Find_cutoff=find(f_slice>=fmin,1);
%calc k
omega=f_slice*2*pi;
L0=g*f_slice.^(-2)/(2*pi);
k_guess=2*pi*L0.^(-1);
pp=1;
for pp=1:length(f_slice)
delta=1;
while delta>10^(-6)
L(pp)=L0(pp)*tanh(k_guess(pp)*water_level);
k(pp)=2*pi/L(pp);
delta=abs(k(pp)^2-k_guess(pp)^2);
k_guess(pp)=k(pp);
end
end
GN_function=cosh(k*(water_level+adcp_offset))./cosh(k*adcp_offset);
vv=2;
for vv=2:Find_cutoff
merged_trans(vv,:)=merged(vv,:)*GN_function(vv);
end
surf_elevation=sum(merged_trans);
surf_elevation_log(bb,:)=surf_elevation;
%locate upcrossings
% surf_elevation=surf_elevation_log(47,:);
for ss =1:length(surf_elevation)-1
if surf_elevation(ss)<0 & surf_elevation(ss+1)>0;
nm_log(ss)=ss;
end
end
Finder=find(nm_log>0);
% find wavecrest and wavetroughs
tt=1;
for tt = 1:length(Finder)-1
crest(tt)=max(surf_elevation(Finder(tt):Finder(tt+1)));
trough(tt)=min(surf_elevation(Finder(tt):Finder(tt+1)));
end
Waveheight=crest-trough;
Hm=mean(Waveheight)
133
sort_fx=sort(Waveheight,'descend');
length_vector=round(length(sort_fx)/3);
Hsig=mean(sort_fx(1:length_vector));
%Log values
Tp(bb)=Tpeak;
Hmean(bb)=Hm;
Hs(bb)=Hsig;
Hmax(bb)=max(Waveheight);
clear Waveheight Hm Finder tt ii crest trough nm_log new_spec Tpeak
end
time_vector=[1:seg_num]*segment/60;
figure
plot(time_vector,Tp,'-.')
title([char('Peak wave period, Offshore ADCP, h=21 m','averaging segment size (in minutes):
',num2str(segment),' Origin:',datestr(time_origo,0),'')])
xlabel('time [hours]');
ylabel('seconds');
%dateaxis('x',13,time_origo)
axis([0 50 0 20])
saveas(gcf,'Tp_ADCP_48hours', 'fig')
Tp=Tp';
save Tp_48hours.txt Tp -ASCII
figure
plot(time_vector,Hmean,'r')
title([char('Wave Height, Offshore ADCP, h=21 m','averaging segment size (in minutes): ',num2str(segment),'
Origin:',datestr(time_origo,0),'')])
hold on
plot(time_vector,Hs,'-.')
hold on
plot(time_vector,Hmax,'-og')
xlabel('time [hours]');
ylabel('meters');
legend('Hmean','Hsig','Hmax')
saveas(gcf,'Waveheight_ADCP_48hours', 'fig')
figure
plot(time_vector,wave_dir,'rx-')
title([char('Wave Direction, Offshore ADCP, h=21 m','averaging segment size (in minutes):
',num2str(segment),' Origin:',datestr(time_origo,0),'')])
xlabel('time [hours]');
ylabel('Direction');
saveas(gcf,'Wavedirection_ADCP_48hours', 'fig')
axis([0 48 310 340]);
figure
plot(time/3600,delta_p)
hold on
title('Pressure timeseries ADCP (tidal variation extracted)')
plot(time/3600,waterdepth-mean(waterdepth),'r--','LineWidth',4)
xlabel('time [hours]');
ylabel('pressure [dBar]');
134
legend('Pressure flucturation due to wave motion','Pressure flucturation due to tidal motion')
axis([0 50 -2 2]);
figure
plot(time/3600,pressure)
title('Pressure timeseries ADCP (Raw)')
xlabel('time [hours]');
ylabel('pressure [dBar]');
axis([0 50 19 23]);
Hmean=Hmean';
Hs=Hs';
Hmax=Hmax';
wave_dir=wave_dir';
save Hm_48hours.txt Hmean -ASCII
save Hsig_48hours.txt Hs -ASCII
save Hmax_48hours.txt Hmax -ASCII
save wavedir_48hours.txt wave_dir -ASCII
toc
135
Appendix IV
Matlab code used to extrat selected velocity and surface elevation time-series from the AquaDopps.
clear all
close all
tic
g=9.81;
rho=1000;
ratio=10^4/(rho*g); %conversion from dbar to pressure in [m]
data06=load('waimea_6m_may06.dat');
pressure=ratio*data06(:,6)
u=data06(:,7);
v=data06(:,8);
time_origo=datenum(2006,04,19,09,00,00);
%Input remarks
adcp_offset=0.30;
dt=1;
time=[0:1:length(pressure)-1]*dt;
%Filtering out the tidal oscilation
average_length=1000;
waterdepth_fraction(1)=mean(pressure(1:average_length));
waterdepth=ones(1,average_length)*waterdepth_fraction(1);
for i=1:fix((length(pressure))/average_length)-1
waterdepth_fraction(i+1)=mean(pressure(i*average_length:(i+1)*average_length));
waterdepth_vector=ones(1,average_length)*waterdepth_fraction(i);
waterdepth=[waterdepth waterdepth_vector];
end
%What is left after the last batch is assigned same waterlevel
av_length=fix((length(pressure))/average_length)*average_length;
rest=length(pressure)-av_length; %last part not included in the averaging
rest_vector=ones(1,rest)*waterdepth_fraction(fix((length(pressure))/average_length));
waterdepth=[waterdepth rest_vector];
delta_p=pressure'-waterdepth;
bins=2^17
% Compute spectrum
fn=1/(dt*2);
Y1=fft(delta_p,bins);
p_spectrum=((real(Y1(1:end)).^2+imag(Y1(1:end)).^2).^0.5)/bins*2;
p_spectrum(1)=p_spectrum(1)/2;
f=[0:bins/2]/bins*2*fn;
Find_max_total=find(p_spectrum==max(p_spectrum),1);
Tpeak_total=(f(Find_max_total))^(-1);
%plot spectrum
figure
plot(f,p_spectrum(1:length(f)))
136
title([char('Wave Powerspectrum (Entire timeseries)',' Tpeak: ',num2str(Tpeak_total),'Starting time:
',num2str(datestr(time_origo,0)),' End time:
',num2str(datestr(time_origo+length(pressure)/(24*3600),0)),'')],'HorizontalAlignment','center');
xlabel('frequency [Hz]');
ylabel('');
saveas(gcf,'Total_spectrum', 'jpeg')
%plotting figures
figure
plot(time,pressure,'r')
hold on
plot(time, waterdepth,'b')
title(['Starting time: ',num2str(datestr(time_origo,0)),' End time:
',num2str(datestr(time_origo+length(pressure)/(24*3600),0)),'']);
xlabel('time [seconds]');
ylabel('meters');
legend('ADCP pressure', 'Tidal motion');
saveas(gcf,'ADCP_pressure', 'jpeg')
%............................................%
% Selecting the surface elevation timeseries %
%............................................%
%Slice up the selected surface elevation file
begin=83700+4200;
duration=30*60;
stop=begin+duration;
time_slice=time(begin:stop);
p_slice=delta_p(begin:stop)';
relative_t=[0:dt:length(p_slice)-1];
%Extract velocities
u_slice_thursday_925=u(begin:stop);
v_slice_thursday_925=v(begin:stop);
fitfunction=fit(u_slice_thursday_925,v_slice_thursday_925,'poly1');
Mag_correction=10.4; % correction for magnetic north
wave_dir=abs(atan2(1,fitfunction.p1)/pi*180)+180+Mag_correction;
figure
plot(u_slice_thursday_925,v_slice_thursday_925,'.')
hold on
title(['AQDP06 wave direction: ',num2str(wave_dir),' degrees'])
plot(u_slice_thursday_925,fitfunction.p1*u_slice_thursday_925,'r')
axis([-1.5 1.5 -1.5 1.5])
save u_AQDP06_thursday_925.mat u_slice_thursday_925 -MAT
save v_AQDP06_thursday_925.mat v_slice_thursday_925 -MAT
% save pressure file
save pressure_AQDP06_thursday925_v2.txt p_slice -ASCII
save pressure_AQDP06_thursday925_v2.mat p_slice -mat
% Compute spectrum of selected slice
bins_slice=2^12;
Y=fft(p_slice,bins_slice);
137
p_slice_spectrum=((real(Y(1:end)).^2+imag(Y(1:end)).^2).^0.5)/bins*2;
p_slice_spectrum(1)=p_slice_spectrum(1)/2;
f_slice=[0:bins_slice/2]/bins_slice*2*fn;
Find_max=find(p_slice_spectrum==max(p_slice_spectrum),1);
Tpeak=(f_slice(Find_max))^(-1);
water_level=mean(waterdepth(begin:stop))+adcp_offset;
% Plot spectrum of selected slice
figure
plot(f_slice,p_slice_spectrum(1:length(f_slice)))
title([char('Wave Powerspectrum (Selected timeseries) Tpeak: ',num2str(Tpeak),'Starting time:
',num2str(datestr(time_origo+begin/(24*3600),0)),' End time:
',num2str(datestr(time_origo+stop/(24*3600),0)),' ')]);
xlabel('frequency [Hz]');
ylabel('');
saveas(gcf,'wavespectrum_slice', 'jpeg')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Syntesize Untransformed Wavespectrum: Frequency space => Time Series
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Re_Y1=real(Y(1))/bins_slice;
Re_Y2=real(Y(2:end-1))/(bins_slice/2); %Correcting the real parts.
Re_Y3=real(Y(bins_slice/2))/bins_slice;
Re_Y=[Re_Y1' Re_Y2' Re_Y3'];
Im_Y=-imag(Y(1:end))/(bins_slice/2);
f_cutoff=find(f_slice>=0.22,1)%0.337,1);
for kk=1:f_cutoff
merged(kk,:)=Re_Y(kk).*cos(2*pi*f_slice(kk)*relative_t)+Im_Y(kk)*sin(2*pi*f_slice(kk)*relative_t);
end
reform_wave=sum(merged);
% CONTROLING CORRECT FOURIER ANALYSIS
figure
plot(time_slice,p_slice,'r')
title('Original signal compared to fourier synthesis');
hold on
plot(time_slice,reform_wave+2.5,'b')
saveas(gcf,'slice_control', 'jpeg')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Syntesize The transformed Wavespectrum: Frequency space => Time Series
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%calc k
omega=f_slice*2*pi;
L0=g*f_slice.^(-2)/(2*pi);
k_guess=2*pi*L0.^(-1);
for pp=1:length(f_slice)
delta=1;
while delta>10^(-6)
L(pp)=L0(pp)*tanh(k_guess(pp)*water_level);
138
k(pp)=2*pi/L(pp);
delta=abs(k(pp)^2-k_guess(pp)^2);
k_guess(pp)=k(pp);
end
end
GN_function=cosh(k*water_level)./cosh(k*adcp_offset);
figure
plot(k,GN_function)
title('The "pressure=>surface elevation transfer function');
xlabel('k');
ylabel('GN');
saveas(gcf,'GN_function', 'fig')
%plot(k)
merged_trans(1,:)=merged(1,:);
for vv=2:f_cutoff
merged_trans(vv,:)=merged(vv,:)*GN_function(vv);
end
surf_elevation=sum(merged_trans);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure
plot(surf_elevation,'-.')
hold on
plot(p_slice,'r')
title(['Starting time: ',num2str(datestr(time_origo+begin/(24*3600),0)),' End time:
',num2str(datestr(time_origo+stop/(24*3600),0)),'']);
xlabel('time [seconds]');
ylabel('meters');
legend('Surface elevation f = 0 - 0.15 Hz', 'Pressure (ADCP)');
saveas(gcf,'surfelevation_slice', 'fig')
save surf_slice.mat surf_elevation
disp(['Starting time: ',num2str(datestr(time_origo+begin/(24*3600),0)),''])
disp(['Ending time: ',num2str(datestr(time_origo+stop/(24*3600),0)),''])
disp(['Number of timesteps: ',num2str(length(p_slice)),''])
disp(['Average Waterlevel: ',num2str(water_level),'meters'])
disp(['Waterlevel diff: ',num2str(waterdepth(begin)-waterdepth(stop)),'meter'])
disp(['Wave peak period: ',num2str(Tpeak),'sec'])
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
surf_elevation=surf_elevation'
save etha_AQDP06_thursday925_v_cutoff022.txt surf_elevation -ASCII
139
Appendix V
Mablab code used for conducting time-series analysis of model results and calculate Hsig, Hm, Um,
Vm, Umax /Umin and Vmax/Vmin.
close all
clear all
phase_shift=69;
time_length=1799;
Magnetic_North=10.4;
% Load Data from AQDP 06
AQDP06=load('etha_AQDP06_thursday925_v3.txt');
AQDP06_u_load=load('-mat','u_AQDP06_thursday_925.mat');
AQDP06_v_load=load('-mat','v_AQDP06_thursday_925.mat');
AQDP06_u=AQDP06_u_load.u_slice_thursday_925;
AQDP06_v=AQDP06_v_load.v_slice_thursday_925;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Load Data from AQDP 10
AQDP10=load('etha_AQDP10_thursday925_v3.txt')';
AQDP10_u_load=load('-mat','u_AQDP10_thursday_925.mat');
AQDP10_v_load=load('-mat','v_AQDP10_thursday_925.mat');
AQDP10_u=AQDP10_u_load.u_slice_thursday_925;
AQDP10_v=AQDP10_v_load.v_slice_thursday_925;
%%%%%%%%%%%%%%%%%%%%%%%%%
% Load ADCP Velocities
ADCP_u_load=load('-mat','u_slice_thursday925.m');
ADCP_v_load=load('-mat','v_slice_thursday925.m');
ADCP_u=ADCP_u_load.u_velocity_slice./1.75;
ADCP_v=ADCP_v_load.v_velocity_slice./1.75;
%¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤
% Velocity Rotation
%%%%%%%%%%%%%%%%%%%%%%%
rotation=(30-Magnetic_North)*pi/180;
AQDP06_u_rot = AQDP06_u * cos(rotation) - AQDP06_v * sin(rotation);
AQDP06_v_rot = AQDP06_v * cos(rotation) + AQDP06_u * sin(rotation);
AQDP10_u_rot = AQDP10_u * cos(rotation) - AQDP10_v * sin(rotation);
AQDP10_v_rot = AQDP10_v * cos(rotation) + AQDP10_u * sin(rotation);
ADCP_u_rot = ADCP_u * cos(rotation) - ADCP_v * sin(rotation);
ADCP_v_rot = ADCP_v * cos(rotation) + ADCP_u * sin(rotation);
%Version 28
Mike_input_v6=load('ADCPinput_raw_v4.txt');
Mike_offshore_v6=load('etha_offshore_MIKE21_v28.txt');
140
Mike_bayentry_v6=load('etha_bayentry_MIKE21_v28.txt');
Mike_baymiddle_v6=load('etha_baymiddle_MIKE21_v28.txt');
%Version 31
Mike_input_v7=load('ADCPinput_raw_v4.txt');
Mike_offshore_v7=load('etha_offshore_MIKE21_v31.txt');
Mike_bayentry_v7=load('etha_bayentry_MIKE21_v31.txt');
Mike_baymiddle_v7=load('etha_baymiddle_MIKE21_v31.txt');
% Load Mike21 Velocities
h_input=20.98;
Mike_input_u_v6=load('pflux_offshore_MIKE21_v28.txt')/h_input;
Mike_input_v_v6=load('qflux_offshore_MIKE21_v28.txt')/h_input;
h10=10.8;
Mike_bayentry_u_v6=load('pflux_bayentry_MIKE21_v28.txt')./(Mike_bayentry_v6+h10);
Mike_bayentry_v_v6=load('qflux_bayentry_MIKE21_v28.txt')./(Mike_bayentry_v6+h10);
h06=7.4;
Mike_baymiddle_u_v6=load('pflux_baymiddle_MIKE21_v28.txt')./(Mike_baymiddle_v6+h06);
Mike_baymiddle_v_v6=load('qflux_baymiddle_MIKE21_v28.txt')./(Mike_baymiddle_v6+h06);
Mike_input_u_v7=load('pflux_offshore_MIKE21_v31.txt')./(h_input+Mike_input_v7(1:time_length));
Mike_input_v_v7=load('qflux_offshore_MIKE21_v31.txt')./(h_input+Mike_input_v7(1:time_length));
Mike_bayentry_u_v7=load('pflux_bayentry_MIKE21_v31.txt')./(h10+Mike_bayentry_v7+1);
Mike_bayentry_v_v7=load('qflux_bayentry_MIKE21_v31.txt')./(h10+Mike_bayentry_v7+1);
Mike_baymiddle_u_v7=load('pflux_baymiddle_MIKE21_v31.txt')./(h06+Mike_baymiddle_v7+1);
Mike_baymiddle_v_v7=load('qflux_baymiddle_MIKE21_v31.txt')./(h06+Mike_baymiddle_v7+1);
%Intercorrelation between MIKE21 Output
correlation1_v6=lagcor(Mike_input_v6(1:time_length),Mike_offshore_v6(1:time_length),220);
correlation1_v7=lagcor(Mike_input_v7(1:time_length),Mike_offshore_v7(1:time_length),220);
t_vector=[1:time_length]-1;
x_vector=[1:length(correlation1_v7)]-220;
%Plot inputwaveseries vs directional wave series
figure
plot(x_vector,correlation1_v6,'b')
title(char('Correlation between input wave vs resultant directional ','wave at deployment location of ADCP'))
hold on
plot(x_vector,correlation1_v7,'r')
legend('ver 6','ver 11')
xlabel('seconds')
figure
plot(t_vector,Mike_input_v6(1:time_length))
hold on
title('\eta calculated from p^+ ver6 (frequency cutoff = 0.19 Hz)')
plot(t_vector-69,Mike_offshore_v6(1:time_length),'r')
legend('Input timeseries from ADCP','Mike21 prediction at ADCP location')
axis([-70 70 -2.5 2.5])
xlabel('Model run-time (seconds)')
141
ylabel('surface elevation (m)')
figure
plot(t_vector,Mike_input_v7(1:time_length))
hold on
title('\eta calculated from p^+ ver11 (frequency cutoff = 0.19 Hz)')
plot(t_vector-66,Mike_offshore_v7(1:time_length),'r')
legend('Input timeseries from ADCP','Mike21 prediction at ADCP location')
axis([-70 70 -2.5 2.5])
xlabel('Model run-time (seconds)')
ylabel('surface elevation (m)')
figure
plot(t_vector-69,Mike_offshore_v6(1:time_length),'r')
hold on
title('wave-input comparison')
plot(t_vector-66,Mike_offshore_v7(1:time_length),'g')
legend('ver6','ver11 (corrected for reflection)')
axis([-70 70 -2.5 2.5])
xlabel('Model run-time (seconds)')
ylabel('surface elevation (m)')
correlation7_v7=lagcor(Mike_bayentry_v7(1:time_length),AQDP10(1:time_length),220);
correlation7_v6=lagcor(Mike_bayentry_v6(1:time_length),AQDP10(1:time_length),220);
figure
plot(x_vector,correlation7_v6,'r')
hold on
plot(x_vector,correlation7_v7,'g')
title(char('Correlation between Mike21 ver 11 vs field ',' measurements (from Aquadopp) in the bay entry'))
xlabel('seconds')
xlabel('seconds')
legend('ver6','ver9')
figure
plot(t_vector,AQDP10(1:time_length),'k')
hold on
title('\eta calculated from p^+ (frequency cutoff = 0.28 Hz)')
plot(t_vector-70,Mike_bayentry_v6(1:time_length),'b')
hold on
plot(t_vector-72,Mike_bayentry_v7(1:time_length),'r')
legend('AquaDopp placed at Bay Entry (h = 10 m)','Mike21s prediction at same location (ver. 6)','Mike21s
prediction at same location (ver. 11)')
axis([-70 70 -1.5 1.5])
xlabel('Model run-time (seconds)')
ylabel('surface elevation (m)')
correlation8_v6=lagcor(Mike_baymiddle_v6(1:time_length),AQDP06(1:time_length),220);
correlation8_v7=lagcor(Mike_baymiddle_v7(1:time_length),AQDP06(1:time_length),220);
figure
plot(x_vector,correlation8_v6,'b')
hold on
plot(x_vector,correlation8_v7,'r')
title(char('Correlation between Mike21 vs field ',' measurements (from Aquadopp) at the bay middle'))
xlabel('seconds')
xlabel('seconds')
142
legend('ver6','ver 9')
figure
plot(t_vector,AQDP06(1:time_length),'k')
hold on
title('\eta calculated from p^+ (frequency cutoff = 0.34 Hz)')
plot(t_vector-48,Mike_baymiddle_v6(1:time_length),'b')
hold on
plot(t_vector-48,Mike_baymiddle_v7(1:time_length),'r')
legend('AquaDopp placed in the Bay middle (h = 6.8 m)','Mike21s prediction at same location (ver.
6)','Mike21s prediction at same location (ver. 11)')
axis([-70 70 -2.5 2.5])
xlabel('Model run-time (seconds)')
ylabel('surface elevation (m)')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Bayentry velocity comparison
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure
plot(t_vector,AQDP10_u_rot(1:time_length),'k')
title('UO comparison, Bay Entry')
hold on
plot(t_vector-69,Mike_bayentry_u_v6(1:time_length),'b')
hold on
plot(t_vector-66,Mike_bayentry_u_v7(1:time_length),'r')
legend('AquaDopp (h = 10.8 m)','Mike21s prediction at same location (ver. 6)','Mike21s prediction at same
location (ver. 11)')
xlabel('Model run-time (seconds)')
axis([-70 70 -2.5 2.5])
ylabel('velocity (m/s)')
figure
plot(t_vector,AQDP10_v_rot(1:time_length),'k')
title('VO comparison, Bay Entry')
hold on
plot(t_vector-69,Mike_bayentry_v_v6(1:time_length),'b')
hold on
plot(t_vector-66,Mike_bayentry_v_v7(1:time_length),'r')
legend('AquaDopp (h = 10.8 m)','Mike21s prediction at same location (ver. 6)','Mike21s prediction at same
location (ver. 11)')
xlabel('Model run-time (seconds)')
axis([-70 70 -2.5 2.5])
ylabel('velocity (m/s)')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Baymiddle velocity comparison
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure
plot(t_vector,AQDP06_u_rot(1:time_length),'k')
title('UO comparison, Bay middle')
hold on
plot(t_vector-69,Mike_baymiddle_u_v6(1:time_length),'b')
hold on
plot(t_vector-48,Mike_baymiddle_u_v7(1:time_length),'r')
legend('AquaDopp (h = 7.8 m)','Mike21s prediction at same location (ver. 6)','Mike21s prediction at same
location (ver. 11)')
xlabel('Model run-time (seconds)')
axis([-70 70 -2.5 2.5])
143
ylabel('velocity (m/s)')
figure
plot(t_vector,AQDP06_v_rot(1:time_length),'k')
title('VO comparison, Bay middle')
hold on
plot(t_vector-69,Mike_baymiddle_v_v6(1:time_length),'b')
hold on
plot(t_vector-48,Mike_baymiddle_v_v7(1:time_length),'r')
legend('AquaDopp (h = 7.8 m)','Mike21s prediction at same location (ver. 6)','Mike21s prediction at same
location (ver. 11)')
xlabel('Model run-time (seconds)')
axis([-70 70 -2.5 2.5])
ylabel('velocity (m/s)')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Wave direction comparison
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure
% hold on
% plot(Mike_input_u_v7,fitfunction1.p1*Mike_input_u_v7,'r')
% wave_dir(1)=abs(atan2(1,fitfunction1.p1)/pi*180)+210;
plot(Mike_input_u_v6,Mike_input_v_v6,'.g')
hold on
plot(Mike_input_u_v7,Mike_input_v_v7,'xr')
title('Velocity Scatter plot (Station 1, Offshore)')
fitfunction1=fit(-Mike_input_u_v7,Mike_input_v_v7,'poly1');
axis([-2 2 -2 2])
hold on
plot(ADCP_u_rot,ADCP_v_rot,'k+')
legend('Simulation 1','Simulation 2','Field Data')
xlabel('U-velocity [m/s]')
ylabel('V-velocity [m/s]')
% fitfunction2=fit(ADCP_u_rot,ADCP_v_rot,'poly1');
% wave_dir(2)=abs(atan2(1,fitfunction2.p1)/pi*180)+210;
figure
plot(Mike_bayentry_u_v7,Mike_bayentry_v_v7,'x')
axis([-2 2 -2 2])
title('velocity direction, Mike21 Bay Entry ver 7')
fitfunction3=fit(Mike_bayentry_u_v7,Mike_bayentry_v_v7,'poly1');
wave_dir(3)=abs(atan2(1,fitfunction3.p1)/pi*180)+210;
hold on
plot(AQDP10_u_rot,AQDP10_v_rot,'xr')
fitfunction4=fit(AQDP10_u_rot,AQDP10_v_rot,'poly1');
wave_dir(4)=abs(atan2(1,fitfunction4.p1)/pi*180)+210;
legend('v24','AQDP10')
figure
plot(Mike_baymiddle_u_v7,Mike_baymiddle_v_v7,'x')
title('velocity direction, Mike21 Baymiddle ver 17')
axis([-2 2 -2 2])
fitfunction5=fit(Mike_baymiddle_u_v7,Mike_baymiddle_v_v7,'poly1');
wave_dir(5)=abs(atan2(1,fitfunction5.p1)/pi*180)+210;
hold on
plot(AQDP06_u_rot,AQDP06_v_rot,'xr')
144
fitfunction6=fit(AQDP06_u_rot,AQDP06_v_rot,'poly1');
wave_dir(6)=abs(atan2(1,fitfunction6.p1)/pi*180)+210;
legend('v24','AQDP06')
figure
plot(Mike_baymiddle_u_v6,Mike_baymiddle_v_v6,'.g')
title('velocity direction, Mike21 Baymiddle ver 17')
axis([-2 2 -2 2])
hold on
plot(Mike_baymiddle_u_v7,Mike_baymiddle_v_v7,'xr')
title('Velocity Scatter Plot (Station 3, Bay Middle)')
hold on
fitfunction5=fit(Mike_baymiddle_u_v6,Mike_baymiddle_v_v6,'poly1');
wave_dir(5)=abs(atan2(1,fitfunction5.p1)/pi*180)+210;
plot(AQDP06_u_rot,AQDP06_v_rot,'+k')
fitfunction6=fit(AQDP06_u_rot,AQDP06_v_rot,'poly1');
wave_dir(6)=abs(atan2(1,fitfunction6.p1)/pi*180)+210;
legend('Simulation 2','Simulation 3','Field Data')
xlabel('U-velocity [m/s]')
ylabel('V-velocity [m/s]')
figure
plot(Mike_bayentry_u_v6,Mike_bayentry_v_v6,'.g')
title('velocity direction, Mike21 Baymiddle ver 17')
axis([-2 2 -2 2])
hold on
plot(Mike_bayentry_u_v7,Mike_bayentry_v_v7,'xr')
title('Velocity Scatter Plot (Station 2, Bay Entry)')
hold on
fitfunction5=fit(Mike_bayentry_u_v6,Mike_bayentry_v_v6,'poly1');
wave_dir(5)=abs(atan2(1,fitfunction5.p1)/pi*180)+210;
plot(AQDP10_u_rot,AQDP10_v_rot,'+k')
fitfunction6=fit(AQDP06_u_rot,AQDP06_v_rot,'poly1');
wave_dir(6)=abs(atan2(1,fitfunction6.p1)/pi*180)+210;
legend('Simulation 2','Simulation 3','Field Data')
xlabel('U-velocity [m/s]')
ylabel('V-velocity [m/s]')
%% PRESENTATION
figure
plot(t_vector,Mike_input_v7(1:time_length),'k-x','Markersize',6)
hold on
title('Surface elevation comparison (Station 1, Offshore)')
plot(t_vector-73,Mike_offshore_v6(1:time_length),'b--')
hold on
plot(t_vector-73,Mike_offshore_v7(1:time_length),'r')
legend('Field Data','Simulation 2','Simulation 3')
axis([-70 70 -2.5 2.5])
xlabel('Model run-time (seconds)')
ylabel('surface elevation (m)')
figure
plot(t_vector,AQDP10(1:time_length),'k-x','Markersize',5)
hold on
title('Surface elevation Bay Entry (Station 2, Bay Entry)')
% plot(t_vector-70,Mike_bayentry_v6(1:time_length),'b')
% hold on
145
plot(t_vector-73,Mike_bayentry_v6(1:time_length),'b--')
hold on
plot(t_vector-73,Mike_bayentry_v7(1:time_length),'r')
legend('Field Data','Simulation 2','Simulation 3')
axis([-70 70 -2.5 2.5])
xlabel('Model run-time (seconds)')
ylabel('surface elevation (m)')
figure
plot(t_vector,AQDP06(1:time_length),'k-x','Markersize',5)
hold on
title('Surface elevation Bay Middle (Station 3, Bay Middle)')
% plot(t_vector-70,Mike_bayentry_v6(1:time_length),'b')
% hold on
plot(t_vector-73,Mike_baymiddle_v6(1:time_length),'b--')
hold on
plot(t_vector-73,Mike_baymiddle_v7(1:time_length),'r')
legend('Field Data','Simulation 2','Simulation 3')
axis([-70 70 -2.5 2.5])
xlabel('Model run-time (seconds)')
ylabel('surface elevation (m)')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Input u velocity comparison
figure
plot(t_vector,ADCP_u_rot(1:time_length),'k')
title('offshore u-velocity comparison')
hold on
plot(t_vector-69,Mike_input_u_v6(1:time_length),'b')
hold on
plot(t_vector-66,Mike_input_u_v7(1:time_length),'r')
legend('ADCP','ver 6','ver 11')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Input v velocity comparison
figure
plot(t_vector,ADCP_v_rot(1:time_length),'k')
title('offshore v-velocity comparison')
hold on
plot(t_vector-66,Mike_input_v_v7(1:time_length),'r')
legend('ADCP','ver 11')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Wave statistics
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
extra=40;
Surf_matrix=[...
AQDP06(extra:time_length-phase_shift)...
AQDP10(extra:time_length-phase_shift)...
Mike_bayentry_v6(phase_shift+extra:time_length)...
Mike_bayentry_v7(phase_shift+extra:time_length)...
Mike_baymiddle_v6(phase_shift+extra:time_length)...
Mike_baymiddle_v7(phase_shift+extra:time_length)...
Mike_input_v6(phase_shift+extra:time_length)...
146
Mike_input_v7(phase_shift+extra:time_length)...
Mike_offshore_v6(phase_shift+extra:time_length)...
Mike_offshore_v7(phase_shift+extra:time_length)...
];
for ff=1:size(Surf_matrix,2)
surf_elevation=Surf_matrix(:,ff);
%locate upcrossings
for ii =1:length(surf_elevation)-1
if surf_elevation(ii)<0 & surf_elevation(ii+1)>0;
log(ii)=ii;
end
end
Finder=find(log>0);
% find wavecrest and wavetroughs
for kk = 1:length(Finder)-1
crest(kk)=max(surf_elevation(Finder(kk):Finder(kk+1)));
trough(kk)=min(surf_elevation(Finder(kk):Finder(kk+1)));
end
Waveheight=crest-trough;
Hm(ff)=mean(Waveheight);
sort_fx=sort(Waveheight,'descend');
length_vector=round(length(sort_fx)/3);
Hsig(ff)=mean(sort_fx(1:length_vector));
clear Finder Waveheight ii kk crest trough nm_log surf_elevation log
end
%Wave direction Comparison
%Mean waveheight Comparison
disp(['***********************************'])
disp(['* Mean wave height
*'])
disp(['***********************************'])
disp(['ADCP, Hmean = ',num2str(Hm(7),3), ' m'])
disp(['Offshore v6, Hmean = ',num2str(Hm(9),3), ' m'])
disp(['Offshore v22, Hmean = ',num2str(Hm(10),3), ' m'])
disp([' '])
disp(['AQDP10, Hmean = ',num2str(Hm(2),2), ' m'])
disp(['Bayentry Mike21 v6, Hmean = ',num2str(Hm(3),2), ' m'])
disp(['Bayentry Mike21 v22, Hmean = ',num2str(Hm(4),2), ' m'])
disp([''])
disp(['AQDP06, Hmean = ',num2str(Hm(1),3), ' m'])
disp(['Baymiddle Mike21 v6, Hmean = ',num2str(Hm(5),3), ' m'])
disp(['Baymiddle Mike21 v22, Hmean = ',num2str(Hm(6),3), ' m'])
disp([' '])
disp(['************************************'])
disp(['* Significant wave height
*'])
disp(['************************************'])
%Sig waveheight Comparison
disp(['ADCP, Hsig = ',num2str(Hsig(7),3), ' m'])
disp(['Offshore v6, Hsig = ',num2str(Hsig(9),3), ' m'])
disp(['Offshore v22, Hsig = ',num2str(Hsig(10),3), ' m'])
disp([' '])
disp(['AQDP10, Hsig = ',num2str(Hsig(2),3), ' m'])
147
disp(['Bayentry Mike21 v6, Hsig = ',num2str(Hsig(3),3), ' m'])
disp(['Bayentry Mike21 v22, Hsig = ',num2str(Hsig(4),3), ' m'])
disp([' '])
disp(['AQDP06, Hsig = ',num2str(Hsig(1),3), ' m'])
disp(['Baymiddle Mike21 v6, Hsig = ',num2str(Hsig(5),3), ' m'])
disp(['Baymiddle Mike21 v22, Hsig = ',num2str(Hsig(6),3), ' m'])
disp([' '])
disp(['************************************'])
disp(['* Mean Velocities Bay Entry
*'])
disp(['************************************'])
disp(['Um_{AQDP} =', num2str(mean(AQDP10_u_rot),1),'m/s'])
disp(['Um_{Mike21} v6=', num2str(mean(Mike_bayentry_u_v6),1),'m/s'])
disp(['Um_{Mike21} v11=', num2str(mean(Mike_bayentry_u_v7),1),'m/s'])
disp([' '])
disp(['Vm_{AQDP} =', num2str(mean(AQDP10_v_rot),1),'m/s'])
disp(['Vm_{Mike21} v6=', num2str(mean(Mike_bayentry_v_v6),1),'m/s'])
disp(['Vm_{Mike21} v11=', num2str(mean(Mike_bayentry_v_v7),1),'m/s'])
disp([' '])
disp(['************************************'])
disp(['* Max/Min Velocities Bay Entry *'])
disp(['************************************'])
disp(['Umax_{AQDP} =', num2str(max(AQDP10_u_rot),3),'m/s'])
disp(['Umin_{AQDP} =', num2str(min(AQDP10_u_rot),2),'m/s'])
disp([' '])
disp(['Umax_{Mike21} v6 =', num2str(max(Mike_bayentry_u_v6),3),'m/s'])
disp(['Umin_{Mike21} v6=', num2str(min(Mike_bayentry_u_v6),2),'m/s'])
disp([' '])
disp(['Umax_{Mike21} v11=', num2str(max(Mike_bayentry_u_v7),3),'m/s'])
disp(['Umin_{Mike21} v11=', num2str(min(Mike_bayentry_u_v7),2),'m/s'])
disp([' '])
disp(['Vmax_{AQDP} =', num2str(max(AQDP10_v_rot),2),'m/s'])
disp(['Vmin_{AQDP} =', num2str(min(AQDP10_v_rot),2),'m/s'])
disp([' '])
disp(['Vmax_{Mike21} v6 =', num2str(max(Mike_bayentry_v_v6),3),'m/s'])
disp(['Vmin_{Mike21} v6=', num2str(min(Mike_bayentry_v_v6),3),'m/s'])
disp([' '])
disp(['Vmax_{Mike21} v11=', num2str(max(Mike_bayentry_v_v7),3),'m/s'])
disp(['vmin_{Mike21} v11=', num2str(min(Mike_bayentry_v_v7),2),'m/s'])
disp([' '])
disp(['************************************'])
disp(['* Mean Velocities Bay Middle
*'])
disp(['************************************'])
disp(['Um_{AQDP} =', num2str(mean(AQDP06_u_rot),2),'m/s'])
disp(['Um_{Mike21} v6=', num2str(mean(Mike_baymiddle_u_v6),1),'m/s'])
disp(['Um_{Mike21} v11=', num2str(mean(Mike_baymiddle_u_v7),1),'m/s'])
disp([' '])
disp(['Vm_{AQDP} =', num2str(mean(AQDP06_v_rot),1),'m/s'])
disp(['Vm_{Mike21} v6=', num2str(mean(Mike_baymiddle_v_v6),1),'m/s'])
disp(['Vm_{Mike21} v11=', num2str(mean(Mike_baymiddle_v_v7),1),'m/s'])
disp([' '])
disp(['************************************'])
disp(['* Max/Min Velocities Bay Middle *'])
disp(['************************************'])
disp(['Umax_{AQDP} =', num2str(max(AQDP06_u_rot),3),'m/s'])
disp(['Umin_{AQDP} =', num2str(min(AQDP06_u_rot),3),'m/s'])
disp([' '])
148
disp(['Umax_{Mike21} v6 =', num2str(max(Mike_baymiddle_u_v6),3),'m/s'])
disp(['Umin_{Mike21} v6=', num2str(min(Mike_baymiddle_u_v6),3),'m/s'])
disp([' '])
disp(['Umax_{Mike21} v11=', num2str(max(Mike_baymiddle_u_v7),3),'m/s'])
disp(['Umin_{Mike21} v11=', num2str(min(Mike_baymiddle_u_v7),3),'m/s'])
disp([' '])
disp(['Vmax_{AQDP} =', num2str(max(AQDP06_v_rot),2),'m/s'])
disp(['Vmin_{AQDP} =', num2str(min(AQDP06_v_rot),2),'m/s'])
disp([' '])
disp(['Vmax_{Mike21} v6 =', num2str(max(Mike_baymiddle_v_v6),3),'m/s'])
disp(['Vmin_{Mike21} v6=', num2str(min(Mike_baymiddle_v_v6),3),'m/s'])
disp([' '])
disp(['Vmax_{Mike21} v11=', num2str(max(Mike_baymiddle_v_v7),3),'m/s'])
disp(['Vmin_{Mike21} v11=', num2str(min(Mike_baymiddle_v_v7),3),'m/s'])
disp([' '])
disp(['************************************'])
disp(['* Mean Velocities Offshore
*'])
disp(['************************************'])
disp(['Um_{AQDP} =', num2str(mean(ADCP_u_rot),2),'m/s'])
disp(['Um_{Mike21} v6=', num2str(mean(Mike_input_u_v6),1),'m/s'])
disp(['Um_{Mike21} v11=', num2str(mean(Mike_input_u_v7),1),'m/s'])
disp([' '])
disp(['Vm_{AQDP} =', num2str(mean(ADCP_v_rot),1),'m/s'])
disp(['Vm_{Mike21} v6=', num2str(mean(Mike_input_u_v6),1),'m/s'])
disp(['Vm_{Mike21} v11=', num2str(mean(Mike_input_v_v7),1),'m/s'])
disp([' '])
disp(['************************************'])
disp(['* Max/Min Velocities Offshore *'])
disp(['************************************'])
disp(['Umax_{AQDP} =', num2str(max(ADCP_u_rot),3),'m/s'])
disp(['Umin_{AQDP} =', num2str(min(ADCP_u_rot),3),'m/s'])
disp([' '])
disp(['Umax_{Mike21} v6 =', num2str(max(Mike_input_u_v6),3),'m/s'])
disp(['Umin_{Mike21} v6=', num2str(min(Mike_input_u_v6),3),'m/s'])
disp([' '])
disp(['Umax_{Mike21} v11=', num2str(max(Mike_input_u_v7),3),'m/s'])
disp(['Umin_{Mike21} v11=', num2str(min(Mike_input_u_v7),3),'m/s'])
disp([' '])
disp(['Vmax_{AQDP} =', num2str(max(ADCP_v_rot),2),'m/s'])
disp(['Vmin_{AQDP} =', num2str(min(ADCP_v_rot),2),'m/s'])
disp([' '])
disp(['Vmax_{Mike21} v6 =', num2str(max(Mike_input_v_v6),3),'m/s'])
disp(['Vmin_{Mike21} v6=', num2str(min(Mike_input_v_v6),3),'m/s'])
disp([' '])
disp(['Vmax_{Mike21} v11=', num2str(max(Mike_input_v_v7),3),'m/s'])
disp(['Vmin_{Mike21} v11=', num2str(min(Mike_input_v_v7),3),'m/s'])
disp([' '])
149
Appendix VI
Matlab code used for spectral analysis of model results.
close all
clear all
dt=1;
fn=1/(dt*2);
bins=2^12
f=[1:bins/2]/bins*2*fn;
df=(bins*2*fn)^(-1);
f_band=0.01;
bin_size=round(f_band/df);
bin_number=fix(length(f)/bin_size);
f_new=[0.5:bin_number-0.5]*f_band;
% Load Field data
ADCP=load('ADCPinput_raw_v4.txt');
AQDP10=load('etha_AQDP10_thursday925_v3.txt')';
AQDP06=load('etha_AQDP06_thursday925_v3.txt');
% Load Mike21 Data
Mike_offshore=load('etha_offshore_MIKE21_v28.txt');
Mike_bayentry=load('etha_bayentry_MIKE21_v28.txt');
Mike_baymiddle=load('etha_baymiddle_MIKE21_v28.txt');
l_diff=length(ADCP)-length(Mike_offshore);
phase(1)=73;
phase(2)=73;
phase(3)=73;
Surf_matrix=[...
ADCP(1:end-phase(1)-l_diff+1)...
Mike_offshore(phase(1):end)...
AQDP10(1:end-phase(2)-l_diff+1)...
Mike_bayentry(phase(2):end)...
AQDP06(1:end-phase(3)-l_diff+1)...
Mike_baymiddle(phase(3):end)...
];
%plot(Surf_matrix(:,5:6))
for ii=1:size(Surf_matrix,2)
surf=Surf_matrix(:,ii);
Y=abs(fft(surf,bins));
A=2*Y(2:bins/2+1)./bins;
Power=Y(2:bins/2+1).^2;
Power_scale=Power*2/bins^2/df;
for cc=1:bin_number
new_spect(cc)=mean(Power_scale((cc-1)*bin_size+1:cc*bin_size));
new_A(cc)=mean(A((cc-1)*bin_size+1:cc*bin_size));
end
M0=sum(new_spect*bin_size)/length(AQDP06);
150
Hm0(ii)=4*M0.^0.5;
spect(:,ii)=new_spect;
amp(:,ii)=new_A;
end
thursday=spect(:,1);
figure
subplot(2,2,1),semilogy(f_new,spect(:,1),'-k')
hold on
semilogy(f_new,spect(:,2),'-r')
xlabel('Frequency [Hz]')
ylabel('Power [m^2/Hz]')
title('Offshore (h = 21 m)')
legend('Field Data','Mike21 BW')
save thursday925_spect.txt thursday -ASCII
subplot(2,2,2),semilogy(f_new,spect(:,3),'-k')
hold on
semilogy(f_new,spect(:,4),'-r')
xlabel('Frequency [Hz]')
ylabel('Power [m^2/Hz]')
title('Bay Entry (h = 10.8 m)')
legend('Field Data','Mike21 BW')
subplot(2,2,3),semilogy(f_new,spect(:,5),'-k')
hold on
semilogy(f_new,spect(:,6),'-r')
xlabel('Frequency [Hz]')
ylabel('Power [m^2/Hz]')
title('Bay Middle (h = 7.4 m)')
legend('Field Data','Mike21 BW')
subplot(2,2,4), plot([0 568 683],[Hm0(1) Hm0(3) Hm0(5)],'k-o')
hold on
plot([0 568 683],[Hm0(2) Hm0(4) Hm0(6)],'r-x')
title('Significant waveheight')
xlabel('Distance from generation line [m]')
ylabel('Hm0 [m]')
legend('Field Data','Mike21 BW')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure
subplot(2,2,1),plot(f_new,spect(:,1),'-k')
hold on
plot(f_new,spect(:,2),'-r')
xlabel('Frequency [Hz]')
ylabel('Power [m^2/Hz]')
title('Offshore (h = 21 m)')
legend('Field Data','Mike21 BW')
subplot(2,2,2),plot(f_new,spect(:,3),'-k')
hold on
plot(f_new,spect(:,4),'-r')
xlabel('Frequency [Hz]')
ylabel('Power [m^2/Hz]')
title('Bay Entry (h = 10.8 m)')
151
legend('Field Data','Mike21 BW')
subplot(2,2,3),plot(f_new,spect(:,5),'-k')
hold on
plot(f_new,spect(:,6),'-r')
xlabel('Frequency [Hz]')
ylabel('Power [m^2/Hz]')
title('Bay Middle (h = 7.4 m)')
legend('Field Data','Mike21 BW')
subplot(2,2,4), plot([0 568 683],[Hm0(1) Hm0(3) Hm0(5)],'k-o')
hold on
plot([0 568 683],[Hm0(2) Hm0(4) Hm0(6)],'r-x')
title('Significant waveheight')
xlabel('Distance from generation line [m]')
ylabel('Hm0 [m]')
legend('Field Data','Mike21 BW')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure
set(gcf, 'PaperUnits', 'inches');
set(gcf, 'PaperSize', [50 7]);
subplot(2,2,1),plot(f_new,amp(:,1),'-ok','LineWidth',1,'MarkerSize',5)
hold on
plot(f_new,amp(:,2),'-xr','LineWidth',1)
xlabel('Frequency [Hz]','FontSize',8)
ylabel('Harmonic Amplitude [m]','FontSize',8)
title('Offshore (h = 21 m)')
axis([0 0.5 0 0.06])
legend('Field Data','Mike21 BW')
subplot(2,2,2),plot(f_new,amp(:,3),'-ok','LineWidth',1,'MarkerSize',5)
hold on
plot(f_new,amp(:,4),'-xr','LineWidth',1)
xlabel('Frequency [Hz]','FontSize',8)
ylabel('Harmonic Amplitude [m]','FontSize',8)
title('Bay Entry (h = 10.8 m)')
axis([0 0.5 0 0.06])
legend('Field Data','Mike21 BW')
subplot(2,2,3),plot(f_new,amp(:,5),'-ok','LineWidth',1,'MarkerSize',5)
hold on
plot(f_new,amp(:,6),'-xr','LineWidth',1)
xlabel('Frequency [Hz]','FontSize',8)
ylabel('Harmonic Amplitude [m]','FontSize',8)
title('Bay Middle (h = 7.4 m)')
axis([0 0.5 0 0.06])
legend('Field Data','Mike21 BW')
subplot(2,2,4), plot([0 568 683],[Hm0(1) Hm0(3) Hm0(5)],'k-o','LineWidth',1,'MarkerSize',5)
hold on
plot([0 568 683],[Hm0(2) Hm0(4) Hm0(6)],'r-x','LineWidth',1)
title('Significant Waveheight')
xlabel('Distance from generation line [m]','FontSize',8)
ylabel('Hm0 [m]','FontSize',8)
axis([0 900 0 3.5])
legend('Field Data','Mike21 BW')
152
Appendix VII
Matlab code used to calculate wave height distribution and a filtered envelope plot along the
centerline of Waimea bay.
close all
clear all
dt=1;
fn=1/(dt*2);
bins=2^12
f=[1:bins/2]/bins*2*fn;
df=(bins*2*fn)^(-1);
f_band=0.01;
bin_size=round(f_band/df);
bin_number=fix(length(f)/bin_size);
f_new=[0.5:bin_number-0.5]*f_band;
phase=73;
%%%%%%%%%%%%%%%%%%%%%%%%%
% SURFLINE CALC
%%%%%%%%%%%%%%%%%%%%%%%
Surfline=load('surfline_thursday925_v28.txt');
for ii=1:size(Surfline,2)
ii;
surf=Surfline(:,ii);
Y=abs(fft(surf,bins));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Syntesize Untransformed Wavespectrum: Frequency space => Time Series
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
relative_t=[1:1798];
Re_Y1=real(Y(1))/bins;
Re_Y2=real(Y(2:end-1))/(bins/2); %Correcting the real parts.
Re_Y3=real(Y(bins/2))/bins;
Re_Y=[Re_Y1 Re_Y2' Re_Y3];
Im_Y=-imag(Y(1:end))/(bins/2);
%Create low-pass filter
fmin=0.061;%0.05;
fmax=0.069;%0.06; %Hz
f2=0.083;
f3=0.19;
Find_fmax=find(f>=fmax,1);
Find_fmin=find(f>=fmin,1);
Find_f2=find(f>=f2,1);
Find_f3=find(f>=f3,1);
for kk=Find_fmin:Find_fmax
merged(kk,:)=Re_Y(kk).*cos(2*pi*f(kk)*relative_t)+Im_Y(kk)*sin(2*pi*f(kk)*relative_t);
end
153
for tt=Find_f2:Find_f3
merged2(tt,:)=Re_Y(tt).*cos(2*pi*f(tt)*relative_t)+Im_Y(tt)*sin(2*pi*f(tt)*relative_t);
end
filter_surf(:,ii)=sum(merged);
filter_surf2(:,ii)=sum(merged2);
end
Bat=load('bathymetry_final1.txt');
%%%%%%%%%%%%%%%%%
% Plot figure
%%%%%%%%%%%%%%%%%
wall=0;
x_vector=[1:699]*2;
figure
plot(x_vector-352,Surfline(500:1798,:))
% hold on
% plot(x_vector(1:593),etha,'k','Linewidth',3)
axis([1 900 -2.6 2.6])
hold on
plot(x_vector-352,Bat(1:699),'y','Linewidth',3)
ylabel('Surface elevation [m]')
xlabel('Distance from Wave Generation line [m]')
hold on
plot(x_vector-352,Bat(1:699),'y','Linewidth',3)
figure
plot(x_vector-352,filter_surf(500:1798,:)')
% hold on
% plot(x_vector(1:593)-352,etha,'k','Linewidth',3)
axis([1 900 -0.5 0.5])
hold on
plot(x_vector-352,Bat(1:699),'y','Linewidth',3)
title('Filtered Envelope plot, Time-series 2 ( T=14.4 sec-16.4 sec)')
ylabel('Surface elevation [m]')
xlabel('Distance from Wave Generation line [m]')
figure
plot(x_vector-352,filter_surf2(500:1798,:)')
% hold on
% plot(x_vector(1:593)-352,etha,'k','Linewidth',3)
axis([1 900 -0.5 0.5])
hold on
plot(x_vector-352,Bat(1:699),'y','Linewidth',3)
ylabel('Surface elevation [m]')
xlabel('Distance from Wave Generation line [m]')
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Appendix VIII
3 CD-RWs containing:
CD 1
 3D Animation of Simulation 2.
CD 2
 Time-stamped video clip recorded during time-series 1 (Tape1 Clip2.avi)
 Time-stamped video clip recorded during time-series 2 #1 (Tape2 Clip1 Sub1.avi)
CD 3
 Time-stamped video clip recorded during time-series 2 #2 (Tape2 Clip1 Sub2.avi)
 Time-stamped video clip of the swash zone recorded 30 minutes after time-series 2
(Tape2 Clip2 Sub1.avi)
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